Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\csc (x)$ .

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

Step 1.1.2

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

$- \csc (x) \cot (x)$

Step 1.1.3

Reorder the factors of $- \csc (x) \cot (x)$ .

$- \cot (x) \csc (x)$

Step 1.2

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

$\int - 1 u d u$

Step 2

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

$- \int u d u$

Step 3

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$- (\frac{1}{2} u^{2} + C)$

Step 4

Rewrite $- (\frac{1}{2} u^{2} + C)$ as $- \frac{1}{2} u^{2} + C$ .

$- \frac{1}{2} u^{2} + C$

Step 5

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

$- \frac{1}{2} \csc^{2} (x) + C$