Evaluate the integral: I = integral ((x+1)²)/(sqrt(x)) dx
Evaluate the integral: $$ I = \int \frac{(x+1)^2}{\sqrt{x}} \, dx. $$
Explanation of Method
When a polynomial is divided by a single power of $x$, expanding the polynomial first and then dividing term by term converts the integrand into a sum of pure powers of $x$. Each resulting power can then be integrated directly with the power rule.
- Expand the numerator using standard algebraic identities.
- Divide each term by the denominator, subtracting exponents.
- Integrate each power of $x$ term by term.
- Combine the results and add the constant of integration.
Step 1 · Expand the numerator
The numerator is $(x+1)^2$, which expands as:
$$ (x+1)^2 = x^2 + 2x + 1. $$
Thus, the integrand becomes:
$$ \frac{(x+1)^2}{\sqrt{x}} = \frac{x^2}{\sqrt{x}} + \frac{2x}{\sqrt{x}} + \frac{1}{\sqrt{x}}. $$
Simplify each term:
$$ \frac{x^2}{\sqrt{x}} = x^{3/2}, \qquad \frac{2x}{\sqrt{x}} = 2x^{1/2}, \qquad \frac{1}{\sqrt{x}} = x^{-1/2}. $$
Therefore, the integral becomes:
$$ I = \int x^{3/2}\,dx + 2\int x^{1/2}\,dx + \int x^{-1/2}\,dx. $$
Step 2 · Integrate each term
Using the power rule for integration,
$$ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1, $$
we compute each term.
First term: $\displaystyle \int x^{3/2}\,dx$:
$$ \int x^{3/2}\,dx = \frac{x^{3/2+1}}{3/2+1} = \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2}. $$
Second term: $\displaystyle 2\int x^{1/2}\,dx$:
$$ \int x^{1/2}\,dx = \frac{x^{1/2+1}}{1/2+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}. $$
Multiply by $2$:
$$ 2\int x^{1/2}\,dx = 2\cdot \frac{2}{3}x^{3/2} = \frac{4}{3}x^{3/2}. $$
Third term: $\displaystyle \int x^{-1/2}\,dx$:
$$ \int x^{-1/2}\,dx = \frac{x^{-1/2+1}}{-1/2+1} = \frac{x^{1/2}}{1/2} = 2x^{1/2}. $$
Step 3 · Combine the results
Combine all three terms:
$$ I = \frac{2}{5}x^{5/2} + \frac{4}{3}x^{3/2} + 2x^{1/2} + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Rewritten as | Antiderivative |
|---|---|---|
| $x^2/\sqrt{x}$ | $x^{3/2}$ | $\dfrac{2}{5}x^{5/2}$ |
| $2x/\sqrt{x}$ | $2x^{1/2}$ | $\dfrac{4}{3}x^{3/2}$ |
| $1/\sqrt{x}$ | $x^{-1/2}$ | $2x^{1/2}$ |
Additional Notes
- The domain of $I$ requires $x > 0$, since $\sqrt{x}$ is defined for non-negative $x$ and appears in a denominator.
- Expanding the numerator before dividing is generally faster than trying to integrate $(x+1)^2/\sqrt{x}$ directly as a product or via substitution.
- The result can be checked by differentiating $\dfrac{2}{5}x^{5/2} + \dfrac{4}{3}x^{3/2} + 2x^{1/2}$ and confirming it returns the original integrand.
Result
The solution to the integral is: