Integrals · MathLopedia

Problem 1

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{\sqrt{x} - 2 \sqrt[3]{x^2} + 1}{\sqrt[4]{x}} \, dx. $$

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Problem 2

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1-x^4}} \, dx. $$

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Problem 3

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{(x+1)^2}{\sqrt{x}} \, dx. $$

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Problem 4

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \left(1 - \frac{1}{x^2}\right) \sqrt{x\sqrt{x}} \, dx. $$

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Problem 5

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} \, dx. $$

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Problem 6

Medium Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x^2}{1 - x^2} \, dx. $$

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Problem 7

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x^2 + 3}{x^2 - 1} \, dx. $$

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Problem 8

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{\sqrt{x^2 + 2x + 1}}{x} \, dx. $$

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Problem 9

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x^2}{1 + x^2} \, dx. $$

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Problem 10

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x^2 + 2}{x^2 + 1} \, dx. $$

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Problem 11

Medium Algebraic Integrands

Evaluate the integral: $$ \int \left(3^x - 4^x\right) dx. $$

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Problem 12

Medium Algebraic Integrands

Evaluate the integral: $$ \int 5^x 3^{-x} \, dx. $$

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Problem 13

Hard Algebraic Integrands

Evaluate the integral: $$ \int \left(2^x + 5^x\right)^2 \, dx. $$

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Problem 14

Hard Algebraic Integrands

Evaluate the integral: $$ \int \frac{2^{x+1} - 5^{x-1}}{10^x} \, dx. $$

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Problem 15

Hard Algebraic Integrands

Evaluate the integral: $$ \int \frac{e^{3x} + 1}{e^x + 1} \, dx. $$

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Problem 16

Easy Algebraic Integrands

Evaluate the integral: $$ \int (2x - 3\sin x + \cos x) \, dx. $$

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Problem 17

Hard Algebraic Integrands

Evaluate the integral: $$ \int \sqrt{1 - \sin 2x} \, dx. $$

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Problem 18

Medium Algebraic Integrands

Evaluate the integral: $$ \int \tan^2 x \, dx. $$

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Problem 19

Medium Algebraic Integrands

Evaluate the integral: $$ \int \cot^2 x \, dx. $$

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Problem 20

Medium Algebraic Integrands

Evaluate the integral: $$ \int (m \sinh x + n \cosh x) \, dx, $$ where $m$ and $n$ are constants.

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Problem 21

Hard Algebraic Integrands

Evaluate the integral: $$ \int \tanh^2 x \, dx. $$

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Problem 22

Hard Algebraic Integrands

Evaluate the integral: $$ \int \coth^2 x \, dx. $$

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Problem 23

Hard Algebraic Integrands

Evaluate the integral: $$ \int (5 - 2x)^9 \, dx. $$

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Problem 24

Hard Algebraic Integrands

Evaluate the integral: $$ \int \frac{dx}{(x-2)^3}. $$

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Problem 25

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \sqrt{2x - 3} \, dx. $$

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Problem 26

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \sqrt[5]{1 - x} \, dx. $$

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Problem 27

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2 - 5x}}. $$

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Problem 28

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{(5x - 2)^{5/2}}. $$

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Problem 29

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{\sqrt[5]{1 - 2x + x^2}}{1 - x} \, dx. $$

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Problem 30

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x}{\sqrt{2x + 5}} \, dx. $$

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Problem 31

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int x \sqrt[3]{x-2} \, dx. $$

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Problem 32

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x + 5}{\sqrt[5]{1 - 2x}} \, dx. $$

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Problem 33

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x^2 - 1}{\sqrt[4]{x + 5}} \, dx. $$

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Problem 34

Hard Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{2 + \sqrt[3]{x - 1}}. $$

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Problem 35

Easy Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{x - 2}{x + 3} \, dx. $$

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Problem 36

Medium Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{2x - 3}{1 - x} \, dx. $$

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Problem 37

Medium Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{(1 + x) \sqrt{x}}. $$

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Problem 38

Medium Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{2x^2 + 1}. $$

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Problem 39

Easy Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{x^2 + 9}. $$

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Problem 40

Easy Algebraic Integrands

Evaluate the integral: $$ I = \int \frac{dx}{x^2 + 3}. $$

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Problem 41

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{2 + 3x^2}. $$

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Problem 42

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{2 - 3x^2}. $$

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Problem 43

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{x^2 - 4}}. $$

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Problem 44

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{x^2 + 7}}. $$

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Problem 45

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{3x^2 - 2}}. $$

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Problem 46

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{5 - x^2}}. $$

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Problem 47

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{3 - 4x^2}}. $$

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Problem 48

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2 - 3x^2}}. $$

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Problem 49

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int e^{-\frac{5}{2}x} \, dx. $$

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Problem 50

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int \left(e^{-x} + e^{-2x}\right) \, dx. $$

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Problem 51

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \sin \frac{2}{3}x \, dx. $$

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Problem 52

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int (\sin 5x - \sin 5\alpha)\,dx, $$ where $\alpha$ is a constant.

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Problem 53

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int \sin(2x+3)\,dx. $$

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Problem 54

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int \cos(1-3x)\,dx. $$

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Problem 55

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\cos^2(3x+2)}. $$

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Problem 56

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2\left(2x + \frac{\pi}{4}\right)}. $$

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Problem 57

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{1 + \cos(x)}. $$

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Problem 58

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{1 - \cos(x)}. $$

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Problem 59

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{1 + \sin(x)}. $$

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Problem 60

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int x\left(x^2 - 13\right)^{23} \, dx. $$

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Problem 61

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int\left(3x^2 - 2x + 1\right)\left(x^3 - x^2 + x - 9\right)^7 dx. $$

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Problem 62

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x\,dx}{3 - 2x^2}. $$

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Problem 63

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x\,dx}{(1 + x^2)^2}. $$

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Problem 64

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int x^2 \sqrt{x^3 - 9} \, dx. $$

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Problem 65

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int x^3 \sqrt[3]{1 + x^2} \, dx. $$

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Problem 66

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x\,dx}{\sqrt{1 - x^2}}. $$

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Problem 67

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{(2x+3)\,dx}{x^2 + 3x - 10}. $$

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Problem 68

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{(x+3)\,dx}{x^2 + 6x - 5}. $$

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Problem 69

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x+1}{\sqrt{x^2 + 2x - 9}} \, dx. $$

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Problem 70

Hard Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x\,dx}{(x^2+1) \sqrt{x^2+1}}. $$

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Problem 71

Hard Rational and Quadratic

Evaluate the integral: $$ \int \left(x^2 - 3\right) \sqrt[5]{x^3 - 9x}\,dx. $$

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Problem 72

Medium Rational and Quadratic

Evaluate the integral: $$ \int \frac{x^2\,dx}{\left(8x^3 + 27\right)^{2/3}}. $$

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Problem 73

Medium Rational and Quadratic

Evaluate the integral: $$ \int \frac{x\,dx}{5 + x^4}. $$

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Problem 74

Medium Rational and Quadratic

Evaluate the integral: $$ \int \frac{x^3\,dx}{x^8 - 2}. $$

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Problem 75

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{x^2+1}{x^4+1} \, dx. $$

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Problem 76

Medium Rational and Quadratic

Evaluate the integral: $$ \int x e^{-x^2} \, dx. $$

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Problem 77

Medium Rational and Quadratic

Evaluate the integral: $$ \int \frac{e^{\frac{1}{x}}}{x^2} \, dx. $$

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Problem 78

Medium Rational and Quadratic

Evaluate the integral: $$ \int \frac{e^x \, dx}{e^x + 2}. $$

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Problem 79

Easy Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{e^x+1}{e^x+x} \, dx. $$

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Problem 80

Medium Rational and Quadratic

Evaluate the integral: $$ I = \int \frac{dx}{e^x + e^{-x}}. $$

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Problem 81

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{e^x}{\sqrt{e^x+1}} \, dx. $$

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Problem 82

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{1+e^{2x}}}. $$

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Problem 83

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{e^{x/2} + e^x}. $$

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Problem 84

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\ln x}{x} \, dx. $$

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Problem 85

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\ln^3 x}{x} \, dx. $$

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Problem 86

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{x \ln^5 x}. $$

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Problem 87

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{1}{1-x^2} \ln \frac{1+x}{1-x} \, dx. $$

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Problem 88

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{x \ln x \ln (\ln x)}. $$

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Problem 89

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \sqrt{\frac{\ln \left(x+\sqrt{1+x^2}\right)}{1+x^2}} \, dx. $$

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Problem 90

Easy Exponential and Logarithmic

Evaluate the integral: $$ I = \int \sin \frac{1}{x} \cdot \frac{1}{x^2} \, dx. $$

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Problem 91

Easy Exponential and Logarithmic

Evaluate the integral: $$ I = \int \sin^5 x \cos x \, dx. $$

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Problem 92

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \sin^3 x \cos^3 x \, dx. $$

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Problem 93

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \tan x \, dx. $$

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Problem 94

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \cot x \, dx. $$

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Problem 95

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \cos^5 x \sin^4 x \, dx. $$

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Problem 96

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\cos x + 1}{\sin x + x} \, dx. $$

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Problem 97

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\sin 2x}{\sin^2 x + 3} \, dx. $$

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Problem 98

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\cos^3 x}{\sin x} \, dx. $$

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Problem 99

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2 x + 5 \cos^2 x}. $$

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Problem 100

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\sin x}{\sqrt{2+\cos x}} \, dx. $$

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Problem 101

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\cos x\,dx}{\sqrt[3]{\sin^2 x}}. $$

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Problem 102

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\sin x\,dx}{\sqrt{\cos^3 x}}. $$

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Problem 103

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\sqrt{\tan x}}{\cos^2 x}\,dx. $$

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Problem 104

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2 x\,\sqrt{\cot x}}. $$

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Problem 105

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \cos^5x\,\sqrt{\sin x}\,dx. $$

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Problem 106

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{1-x^2}\,\arcsin x}. $$

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Problem 107

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{(1+x^2)\arctan x}. $$

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Problem 108

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\arctan^3x}{1+x^2}\,dx. $$

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Problem 109

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{\arctan\sqrt{x}}{\sqrt{x}}\cdot\frac{1}{1+x}\,dx. $$

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Problem 110

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{1-x^2}\,\arcsin^5x}. $$

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Problem 111

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{x\sqrt{x^2-1}}. $$

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Problem 112

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\sin x}. $$

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Problem 113

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{\cos x}. $$

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Problem 114

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{x^3}{x+3}\,dx. $$

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Problem 115

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{(2-x)^2}{2-x^2}\,dx. $$

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Problem 116

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{(x-1)(x+3)}. $$

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Problem 117

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{x^2}{(1-x)^{100}}\,dx. $$

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Problem 118

Hard Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{x}{x^4+3x^2+2}\,dx. $$

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Problem 119

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{dx}{1+e^x}. $$

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Problem 120

Medium Exponential and Logarithmic

Evaluate the integral: $$ I = \int \frac{(1+e^x)^2}{1+e^{2x}}\,dx. $$

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Problem 121

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin^2 x\,dx. $$

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Problem 122

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos^2 x\,dx. $$

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Problem 123

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos x\,\sin3x\,dx. $$

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Problem 124

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin3x\,\sin5x\,dx. $$

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Problem 125

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos3x\,\cos4x\,dx. $$

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Problem 126

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin\frac{x}{2}\,\cos\frac{x}{12}\,dx. $$

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Problem 127

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos\frac{x}{2}\,\cos\frac{x}{3}\,dx. $$

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Problem 128

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin x\,\sin(x+\theta)\,dx. $$

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Problem 129

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin\!\left(2x-\frac{\pi}{6}\right)\cos\!\left(3x+\frac{\pi}{4}\right) dx. $$

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Problem 130

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin x\,\sin2x\,\sin3x\,dx. $$

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Problem 131

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin^3 x\,dx. $$

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Problem 132

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos^3 x\,dx. $$

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Problem 133

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin^4 x\,dx. $$

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Problem 134

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos^4 x\,dx. $$

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Problem 135

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin^4x\,\cos^4x\,dx. $$

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Problem 136

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \tan^3 x\,dx. $$

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Problem 137

Hard Trigonometric Integrals

Evaluate the integral: $$ I = \int \cos2x\,\cot^3x\,dx. $$

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Problem 138

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2x\,\cos^2x}. $$

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Problem 139

Hard Trigonometric Integrals

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2x\,\cos x}. $$

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Problem 140

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \frac{dx}{\cos^4x}. $$

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Problem 141

Hard Trigonometric Integrals

Evaluate the integral: $$ I = \int \sin^2 3x\,\sin^3 2x\,dx. $$

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Problem 142

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \ln x\,dx. $$

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Problem 143

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int x\ln x\,dx. $$

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Problem 144

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int x\ln^2x\,dx. $$

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Problem 145

Easy Trigonometric Integrals

Evaluate the integral (for $n\neq-1$): $$ I = \int x^n\ln x\,dx. $$

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Problem 146

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int x\ln\left(x^2-1\right) dx. $$

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Problem 147

Hard Trigonometric Integrals

Evaluate the integral: $$ I = \int \left(x^2+x\right)\ln(x+1)\,dx. $$

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Problem 148

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \ln\left(x+\sqrt{1+x^2}\right) dx. $$

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Problem 149

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \sqrt{x}\,\ln^2x\,dx. $$

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Problem 150

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int \frac{\ln^2x}{x^2}\,dx. $$

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Problem 151

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int x^3\ln^2x\,dx. $$

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Problem 152

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int \frac{\ln x}{x^3}\,dx. $$

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Problem 153

Hard Trigonometric Integrals

Evaluate the integral: $$ I = \int x^2\ln\left(\frac{1-x}{1+x}\right) dx. $$

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Problem 154

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int e^{x+\ln x}\,dx. $$

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Problem 155

Medium Trigonometric Integrals

Evaluate the integral: $$ I = \int x^2e^{-x}\,dx. $$

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Problem 156

Easy Trigonometric Integrals

Evaluate the integral: $$ J = \int x^3e^{-x^2}\,dx. $$

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Problem 157

Medium Trigonometric Integrals

Evaluate the integral: $$ K = \int e^{\sqrt x}\,dx. $$

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Problem 158

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int x\sin x\,dx. $$

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Problem 159

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int x\cos x\,dx. $$

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Problem 160

Easy Trigonometric Integrals

Evaluate the integral: $$ I = \int x\sinh x\,dx. $$

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Problem 161

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int x\cosh x\,dx. $$

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Problem 162

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \sin(\ln x)\,dx. $$

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Problem 163

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \cos(\ln x)\,dx. $$

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Problem 164

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int x^2\sin2x\,dx. $$

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Problem 165

Hard Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int x^4\cos x\,dx. $$

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Problem 166

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{x}{\cos^2x}\,dx. $$

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Problem 167

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{x\cos x}{\sin^3x}\,dx. $$

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Problem 168

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int e^{-x}\sin x\,dx. $$

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Problem 169

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int e^{ax}\sin bx\,dx. $$

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Problem 170

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int e^{ax}\cos bx\,dx. $$

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Problem 171

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int e^{ax}\sin^2x\,dx. $$

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Problem 172

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int e^{ax}\cos^2x\,dx. $$

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Problem 173

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \arcsin x\,dx. $$

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Problem 174

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \arctan x\,dx. $$

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Problem 175

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \arctan\sqrt x\,dx. $$

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Problem 176

Hard Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int x^2\arccos x\,dx. $$

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Problem 177

Hard Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int (\arcsin x)^2\,dx. $$

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Problem 178

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{\arctan e^x}{e^x}\,dx. $$

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Problem 179

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{\arcsin x}{\sqrt{1+x}}\,dx. $$

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Problem 180

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{xe^{\arctan x}}{(1+x^2)^{3/2}}\,dx. $$

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Problem 181

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \sqrt{a^2-x^2}\,dx. $$

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Problem 182

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \sqrt{x^2 \pm a^2}\,dx. $$

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Problem 183

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int x^2\sqrt{a^2+x^2}\,dx. $$

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Problem 184

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{x^2}{(1+x^2)^2}\,dx. $$

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Problem 185

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{(a^2+x^2)^2}. $$

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Problem 186

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{x^5}{\sqrt{1-x^4}}\,dx. $$

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Problem 187

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{x^2+2x}. $$

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Problem 188

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{x^2+3x}. $$

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Problem 189

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{x^2+x+1}. $$

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Problem 190

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{x^2-x+2}. $$

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Problem 191

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{3x^2-2x-1}. $$

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Problem 192

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{3+x-x^2}. $$

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Problem 193

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{x+x^2}}. $$

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Problem 194

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2ax-x^2}}. $$

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Problem 195

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2x^2+3x}}. $$

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Problem 196

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{4x-x^2}}. $$

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Problem 197

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2x^2-6x+5}}. $$

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Problem 198

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{4x^2+4x-3}}. $$

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Problem 199

Medium Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{2x^2-x+2}}. $$

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Problem 200

Easy Hyperbolic Integrals and Integration by Parts

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{4+2x-x^2}}. $$

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Problem 201

Medium Roots and Binomials

Evaluate the integral: $$ I = \int x\sqrt{x^4+2x^2-1}\,dx. $$

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Problem 202

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{1+x-x^2}}. $$

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Problem 203

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{-2x^2+5x-2}}. $$

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Problem 204

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \sqrt{4x^2+x}\,dx. $$

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Problem 205

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \sqrt{x^2-2x-1}\,dx. $$

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Problem 206

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \sqrt{2+x+x^2}\,dx. $$

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Problem 207

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \sqrt{2+x-x^2}\,dx. $$

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Problem 208

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \sqrt{4x^2-2x+1}\,dx. $$

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Problem 209

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x+1}{x^2+x+1}\,dx. $$

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Problem 210

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{2x+7}{x^2+x-2}\,dx. $$

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Problem 211

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{3x+2}{2x^2+x-3}\,dx. $$

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Problem 212

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{5x+2}{x^2+2x+10}\,dx. $$

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Problem 213

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x-4}{x^2-5x+6}\,dx. $$

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Problem 214

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{3-2x}{x^2-x-2}\,dx. $$

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Problem 215

Medium Roots and Binomials

Evaluate the integral (with $\theta$ treated as a constant parameter): $$ I = \int \frac{x\,dx}{x^2-2x\cos\theta+1}. $$

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Problem 216

Hard Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x+1}{\sqrt{x^2+x+1}}\,dx. $$

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Problem 217

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x}{\sqrt{5+x-x^2}}\,dx. $$

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Problem 218

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x}{\sqrt{3-2x-x^2}}\,dx. $$

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Problem 219

Hard Roots and Binomials

Evaluate the integral: $$ I = \int \frac{3x+1}{\sqrt{2x^2-x+1}}\,dx. $$

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Problem 220

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x\,dx}{\sqrt{1-3x^2-2x^4}}. $$

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Problem 221

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^5}{x^6-x^3-2}\,dx. $$

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Problem 222

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x\sqrt{x^2+x+1}}. $$

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Problem 223

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x^2\sqrt{x^2+x-1}}. $$

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Problem 224

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x+x^3}{\sqrt{1+x^2-x^4}}\,dx. $$

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Problem 225

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^5\,dx}{x^6-x^3-2}. $$

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Problem 226

Hard Roots and Binomials

Evaluate the integral: $$ I = \int x\sqrt{x^4+2x^2-1}\,dx. $$

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Problem 227

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{\sin x + 2\cos x + 3}. $$

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Problem 228

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x^2+5x}. $$

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Problem 229

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^2}{x^2-3x+2}\,dx. $$

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Problem 230

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{2x-3}{(x-2)^2}\,dx. $$

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Problem 231

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^3}{x^2+x+1}\,dx. $$

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Problem 232

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^6}{x^2-1}\,dx. $$

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Problem 233

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^3+1}{x^2-3x+2}\,dx. $$

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Problem 234

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{(x+1)^3}{x^2-1}\,dx. $$

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Problem 235

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^3+x^2}{x^2-3x+2}\,dx. $$

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Problem 236

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^4}{x^2+3}\,dx. $$

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Problem 237

Medium Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x^3+1}. $$

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Problem 238

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x^4-1}. $$

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Problem 239

Hard Roots and Binomials

Evaluate the integral: $$ I = \int \frac{dx}{x^4+1}. $$

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Problem 240

Easy Roots and Binomials

Evaluate the integral: $$ I = \int \frac{x^5}{x^3-1}\,dx. $$

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Problem 241

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x^4+3x^2}. $$

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Problem 242

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x(x-1)(x+2)}. $$

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Problem 243

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x+1}{x(x-1)^3}\,dx. $$

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Problem 244

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{(x^2+1)^3}. $$

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Problem 245

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^3+1}{x(x-1)^3}\,dx. $$

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Problem 246

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{2+3x}{(3-x)^3}\,dx. $$

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Problem 247

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^4}{(x+1)^3}\,dx. $$

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Problem 248

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^3}{(x^2-1)^2}\,dx. $$

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Problem 249

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x}{x^3-1}\,dx. $$

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Problem 250

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^2}{1-x^4}\,dx. $$

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Problem 251

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x^2(x^4-1)}. $$

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Problem 252

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x}{x^3-3x+2}\,dx. $$

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Problem 253

Hard Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{(1+x)\left(1+x^2\right)\left(1+x^3\right)}. $$

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Problem 254

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x^2(1+x^2)^2}. $$

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Problem 255

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x(x^3+4)}. $$

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Problem 256

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^5+1}{x^4(x^2+1)}\,dx. $$

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Problem 257

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2-4x+4\right)\left(x^2-4x+5\right)}. $$

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Problem 258

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^2-x+2}{x^4-5x^2+4}\,dx. $$

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Problem 259

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x^4-x^2-2}. $$

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Problem 260

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{x(x^2+1)^2}. $$

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Problem 261

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x}{x^8-1}\,dx. $$

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Problem 262

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^3+1}{x(x^3-8)}\,dx. $$

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Problem 263

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^3+1}{x(x^3-8)}\,dx. $$

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Problem 264

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^8-1}{x\left(x^8+1\right)}\,dx. $$

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Problem 265

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^8-1}{x\left(x^8+1\right)}\,dx. $$

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Problem 266

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{5x-8}{x^3-4x^2+4x}\,dx. $$

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Problem 267

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^5+x^4-2}{x^4-4x^2+4}\,dx. $$

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Problem 268

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^4+1}{x^6+1}\,dx. $$

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Problem 269

Hard Partial Fractions

Evaluate the integral: $$ I = \int \frac{x+1}{x^3\left(x^2+1\right)^2}\,dx. $$

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Problem 270

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^2}{\left(x^2+1\right)^3}\,dx. $$

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Problem 271

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^3+1}{x\left(x^2+x+1\right)^2}\,dx. $$

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Problem 272

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2+x+1\right)^3}. $$

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Problem 273

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^3+x+1\right)^3}. $$

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Problem 274

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^6}{\left(x^2-5\right)^4}\,dx. $$

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Problem 275

Medium Partial Fractions

Evaluate the integral: $$ I = \int \frac{x^4-1}{x\left(x^4-5\right)\left(x^5-5x+1\right)}\,dx. $$

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Problem 276

Easy Partial Fractions

Evaluate the integral, for $n\in\mathbb N$: $$ I = \int \frac{dx}{1+x^{2n}}. $$

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Problem 277

Easy Partial Fractions

Evaluate the integral, for $k\in\mathbb N$: $$ I = \int \frac{dx}{1-x^{2k}}. $$

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Problem 278

Medium Partial Fractions

Evaluate the integral, for $k,n\in\mathbb N$: $$ I = \int \frac{x^n}{1+x^{2k}}\,dx. $$

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Problem 279

Easy Partial Fractions

Evaluate the integral: $$ I = \int x\sqrt[8]{4+3x}\,dx. $$

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Problem 280

Easy Partial Fractions

Evaluate the integral: $$ I = \int \frac{dx}{1+\sqrt x}. $$

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Problem 281

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt x}{1+\sqrt x}\,dx. $$

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Problem 282

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{1+\sqrt{x-1}}. $$

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Problem 283

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt x+1}{\sqrt x-1}\,dx. $$

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Problem 284

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt x}{(x+1)^2}\,dx. $$

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Problem 285

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{x^2\sqrt x}{x+\sqrt x}\,dx. $$

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Problem 286

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt{x+1}+1}{\sqrt{x+1}-1}\,dx. $$

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Problem 287

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt[3]{3x+1}-1}. $$

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Problem 288

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{(x+1)\sqrt{1-x}}. $$

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Problem 289

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\left(\sqrt{x+2}+1\right)\sqrt{\sqrt{x+2}-1}}. $$

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Problem 290

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt{x+1}+2}{(x+1)^2-\sqrt{x+1}}\,dx. $$

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Problem 291

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt x}{1-\sqrt[8]x}\,dx. $$

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Problem 292

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt x\left(\sqrt[3]x+1\right)^2}. $$

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Problem 293

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt[3]x}{\sqrt{\sqrt{x+1}}}\,dx = \int x^{1/3}(x+1)^{-1/4}\,dx. $$

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Problem 294

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt x+\sqrt[3]x}. $$

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Problem 295

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}\,dx. $$

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Problem 296

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{3x+\sqrt[8]{x^2}}. $$

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Problem 297

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{x^2+\sqrt{1+x}}{\sqrt[7]{1+x}}\,dx. $$

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Problem 298

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{1+\sqrt x+\sqrt{1+x}}. $$

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Problem 299

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2-1\right)\sqrt{1-x^2}}. $$

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Problem 300

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2+1\right)\sqrt{x^2-4}}. $$

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Problem 301

Easy Substitutions

Evaluate the integral: $$ I = \int \sqrt{\frac{x+1}{x-1}}\,dx. $$

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Problem 302

Easy Substitutions

Evaluate the integral: $$ I = \int \sqrt{\frac x{2-x}}\,dx. $$

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Problem 303

Easy Substitutions

Evaluate the integral, for constants $a>b$: $$ I = \int \sqrt{\frac{a-x}{x-b}}\,dx. $$

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Problem 305

Easy Substitutions

Evaluate the integral: $$ I = \int x\sqrt{\frac{x-1}{x+1}}\,dx. $$

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Problem 306

Medium Substitutions

Evaluate the integral: $$ I = \int x^3\sqrt{\frac{1+x^2}{1-x^2}}\,dx. $$

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Problem 307

Medium Substitutions

Evaluate the integral: $$ I = \int \frac1{1-2x}\sqrt{\frac{1+2x}{1-2x}}\,dx = \int\frac{\sqrt{1+2x}}{(1-2x)^{3/2}}\,dx. $$

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Problem 308

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}\,dx. $$

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Problem 309

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{x-\sqrt{x^2-1}}. $$

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Problem 310

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{(1+x)\sqrt{1-x-x^2}}. $$

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Problem 311

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{(x+1)\sqrt{x^2+x+1}}. $$

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Problem 312

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{x^2}{\sqrt{1+x+x^2}}\,dx. $$

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Problem 313

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{1-x+x^2}{\sqrt{1+x-x^2}}\,dx. $$

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Problem 314

Easy Substitutions

Evaluate the integral: $$ I = \int x^2\sqrt{4-x^2}\,dx. $$

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Problem 315

Easy Substitutions

Evaluate the integral, for constant $a>0$: $$ I = \int x^4\sqrt{a^2-x^2}\,dx. $$

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Problem 316

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{x^2\sqrt{x^2-x+1}}. $$

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Problem 317

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{x^3}{\sqrt{x^2+2}}\,dx. $$

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Problem 318

Easy Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{x^3\sqrt{x^2+1}}. $$

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Problem 319

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{x^3}{\sqrt{1+2x-x^2}}\,dx. $$

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Problem 320

Medium Substitutions

Evaluate the integral: $$ I = \int \frac{dx}{x\sqrt{\left(1+x+x^2\right)^3}}. $$

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Problem 321

Hard Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{(x-1)^3\sqrt{x^2+3x+1}}. $$

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Problem 322

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{(x+1)^5\sqrt{x^2+2x}}. $$

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Problem 323

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt{1+x^4}}{1-x^4}\,dx. $$

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Problem 324

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \sqrt[3]{x-x^3}\,dx. $$

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Problem 325

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{x^6\sqrt{x^2-1}}. $$

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Problem 326

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^2}{\sqrt{x^6-1}}\,dx. $$

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Problem 327

Easy Advanced Techniques

Evaluate the integral: $$ I = \int x^7\sqrt{1+x^2}\,dx. $$

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Problem 328

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \sqrt[3]{\frac{(x+1)^2}{(x-1)^5}}\,dx = \int (x+1)^{2/3}(x-1)^{-5/3}\,dx. $$

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Problem 329

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt[3]{(x+1)^2(x-2)^2}} = \int (x+1)^{-2/3}(x-2)^{-2/3}\,dx. $$

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Problem 330

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt[6]{(x-3)^2(x+1)^5}} = \int (x-3)^{-1/3}(x+1)^{-5/6}\,dx. $$

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Problem 331

Medium Advanced Techniques

Evaluate the integral (for $x>1$): $$ I = \int \frac{dx}{\sqrt{(x-1)^2(x+2)}}. $$

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Problem 332

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{(1+x)^3(1-x)}}. $$

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Problem 333

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt{x^2+2}}{x^2+1}\,dx. $$

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Problem 334

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{(x^2+1)\sqrt{x^2-1}}. $$

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Problem 335

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{(1+x^2)\sqrt{1-x^2}}. $$

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Problem 336

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x}{(x-1)^2\sqrt{1+2x-x^2}}\,dx. $$

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Problem 337

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x}{\left(x^2-3x+2\right)\sqrt{x^2-4x+3}}\,dx. $$

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Problem 338

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^3}{(1+x)\sqrt{1+2x-x^2}}\,dx. $$

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Problem 339

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt{x^2+x+1}}{(x+1)^2}\,dx. $$

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Problem 340

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2+x+1\right)\sqrt{x^2+x+1}} = \int\frac{dx}{\left(x^2+x+1\right)^{3/2}}. $$

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Problem 341

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\left(4-2x+x^2\right)\sqrt{2+2x-x^2}}. $$

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Problem 342

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\left(x^2+x+1\right)\sqrt{x^2+x-1}}. $$

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Problem 343

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{x+\sqrt{x^2+x+1}}. $$

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Problem 344

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{1+\sqrt{1-2x-x^2}}. $$

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Problem 345

Medium Advanced Techniques

Evaluate the integral (for $x>0$): $$ I = \int \frac{dx}{\left[1+\sqrt{x(1+x)}\right]^2}. $$

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Problem 346

Easy Advanced Techniques

Evaluate the integral: $$ I = \int x\sqrt{x^2-2x+2}\,dx. $$

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Problem 347

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x-\sqrt{x^2+3x+2}}{x+\sqrt{x^2+3x+2}}\,dx. $$

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Problem 348

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{x^2+1}-\sqrt{x^2-1}}. $$

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Problem 349

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt{x(x+1)}}{\sqrt x+\sqrt{x+1}}\,dx. $$

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Problem 350

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^2+1}{x\sqrt{x^2+x^4+1}}\,dx. $$

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Problem 351

Medium Advanced Techniques

Evaluate the integral (for $x>0$): $$ I = \int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}\,dx. $$

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Problem 352

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^2}{\sqrt{1-x^2}}\,dx. $$

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Problem 353

Medium Advanced Techniques

Evaluate the integral (for $x\geq0$): $$ I = \int \sqrt{x^3+x^4}\,dx = \int x^{3/2}\sqrt{1+x}\,dx. $$

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Problem 354

Easy Advanced Techniques

Evaluate the integral (for $0\leq x\leq1$): $$ I = \int \sqrt{x(1-x^2)}\,dx = \int x^{1/2}(1-x^2)^{1/2}\,dx. $$

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Problem 355

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \left(x^2-1\right)^2\sqrt{x^2-1}\,dx = \int\left(x^2-1\right)^{5/2}dx. $$

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Problem 356

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt{1+\sqrt x}}{\sqrt x}\,dx. $$

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Problem 357

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{1+x^4}}. $$

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Problem 358

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^6}{\sqrt{x^2+1}}\,dx. $$

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Problem 359

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{\sqrt x}{\left(1+\sqrt x\right)^2}\,dx. $$

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Problem 360

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\left(1+x^2\right)^{3/2}}. $$

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Problem 361

Easy Advanced Techniques

Evaluate the integral (for $0\leq x\leq\sqrt3$): $$ I = \int \sqrt{3x-x^3}\,dx = \int x^{1/2}\left(3-x^2\right)^{1/2}dx. $$

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Problem 362

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{x^3}{\sqrt{1+2x^2}}\,dx. $$

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Problem 363

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt[3]{1+x^3}}. $$

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Problem 364

Easy Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{x^3\sqrt[8]{2-x^3}} = \int x^{-3}(2-x^3)^{-1/8}\,dx. $$

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Problem 365

Medium Advanced Techniques

Evaluate the integral: $$ I = \int \frac{dx}{x\sqrt[8]{1+x^3}} = \int x^{-1}(1+x^3)^{-1/8}\,dx. $$

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Problem 366

Easy Advanced Techniques

Evaluate the integral (for $x>0$): $$ I = \int \frac{dx}{x\sqrt{1+x^6}}. $$

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