Evaluate the integral: I = integral dx/(sin² x + 5 cos² x)
Evaluate the integral: $$ I = \int \frac{dx}{\sin^2 x + 5 \cos^2 x}. $$
Explanation of Method
Dividing numerator and denominator by $\cos^2x$ converts any rational function of $\sin^2x$ and $\cos^2x$ into a rational function of $\tan x$, since $\sec^2x=1+\tan^2x$. Substituting $u=\tan x$ then reduces the integral to a standard arctangent form.
- Simplify the denominator using $\sin^2x+\cos^2x=1$.
- Divide numerator and denominator by $\cos^2x$ and substitute $u=\tan x$.
- Apply the standard arctangent formula.
- Substitute back and add the constant of integration.
Step 1 · Simplify the denominator
Recall the identity $\sin^2x+\cos^2x=1$. We aim to express the integrand in a form that can be integrated more easily. First, rewrite the denominator:
$$ \sin^2x+5\cos^2x = \sin^2x+\cos^2x+4\cos^2x = 1+4\cos^2x. $$
Thus, the integral becomes:
$$ I = \int \frac{dx}{1+4\cos^2x}. $$
Step 2 · Divide by $\cos^2x$ and substitute $u = \tan x$
Multiplying numerator and denominator by $\sec^2x$ (which is valid since $\sec^2x/\sec^2x=1$):
$$ I = \int \frac{\sec^2x\,dx}{\sec^2x+4\cos^2x\sec^2x} = \int \frac{\sec^2x\,dx}{\sec^2x+4}. $$
Let $u=\tan x$, so $du=\sec^2x\,dx$ and $\sec^2x=1+u^2$. Substituting:
$$ I = \int \frac{du}{(1+u^2)+4} = \int \frac{du}{u^2+5}. $$
Step 3 · Apply the standard result
We recognize the standard form:
$$ \int \frac{du}{a^2+u^2} = \frac{1}{a}\arctan\left(\frac{u}{a}\right). $$
In our case, $a^2=5$, so $a=\sqrt5$. Therefore, the integral becomes:
$$ I = \frac{1}{\sqrt{5}}\arctan\left(\frac{u}{\sqrt{5}}\right) + C. $$
Step 4 · Substitute back
Substituting back $u = \tan x$, we return to the variable $x$:
$$ I = \frac{1}{\sqrt{5}}\arctan\left(\frac{\tan x}{\sqrt{5}}\right) + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Simplified denominator | $1+4\cos^2x$ |
| Substitution | $u = \tan x$ |
| Antiderivative | $\tfrac{1}{\sqrt5}\arctan(u/\sqrt5)$ |
Additional Notes
- The domain of $I$ excludes $x=\tfrac\pi2+k\pi$ for integer $k$, where $\tan x$ (and the multiplication by $\sec^2x$) is undefined.
- Dividing by $\cos^2x$ and substituting $u=\tan x$ is a standard technique for any integral involving only even powers of $\sin x$ and $\cos x$ — it converts the trigonometric integral into a purely algebraic one in $u$.
- The result can be checked by differentiating $\tfrac{1}{\sqrt5}\arctan(\tan x/\sqrt5)$ with the chain rule and confirming it returns $\dfrac{1}{\sin^2x+5\cos^2x}$.
Result
The solution to the integral is: