Calculus Examples

$\int \frac{\csc^{2} (x)}{\cot (x)} d x$

Let $u = \cot (x)$ . Then $d u = - \csc^{2} (x) d x$ , so $- \frac{1}{\csc^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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

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Differentiate $\cot (x)$ .

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

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- \csc^{2} (x)$

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

$\int \frac{- 1}{u} d u$

Split the fraction into multiple fractions.

$\int - \frac{1}{u} d u$

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

$- \int \frac{1}{u} d u$

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- (\ln (| u |) + C)$

Simplify.

$- \ln (| u |) + C$

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

$- \ln (| \cot (x) |) + C$