Calculus Examples

$\int - \csc^{2} (x) \cot^{3} (x) d x$

Step 1

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

$- \int \csc^{2} (x) \cot^{3} (x) d x$

Step 2

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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Step 2.1

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

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Step 2.1.1

Differentiate $\cot (x)$ .

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

Step 2.1.2

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

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

Step 2.2

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

$- \int - 1 u^{3} d u$

Step 3

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

$- - \int u^{3} d u$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$1 \int u^{3} d u$

Step 4.2

Multiply $\int u^{3} d u$ by $1$ .

$\int u^{3} d u$

Step 5

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4} + C$

Step 6

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

$\frac{1}{4} \cot^{4} (x) + C$