Calculus Examples

$\int \frac{\sin (x)}{1 + \cos^{2} (x)} d x$

Step 1

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- d u = \sin (x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\cos (x)$ .

$\frac{d}{d x} [\cos (x)]$

Step 1.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x)$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{- 1}{1 + u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{1}{1 + u^{2}} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{1}{1 + u^{2}} d u$

Step 4

Rewrite $1$ as $1^{2}$ .

$- \int \frac{1}{1^{2} + u^{2}} d u$

Step 5

The integral of $\frac{1}{1^{2} + u^{2}}$ with respect to $u$ is $\arctan (u) + C$ .

$- (\arctan (u) + C)$

Step 6

Simplify.

$- \arctan (u) + C$

Step 7

Replace all occurrences of $u$ with $\cos (x)$ .

$- \arctan (\cos (x)) + C$