Calculus Examples

$\int \frac{\cos^{4} (x)}{\sin^{6} (x)} d x$

Step 1

Simplify.

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Step 1.1

Multiply by $1$ .

$\int \frac{\cos^{4} (x) \cdot 1}{\sin^{6} (x)} d x$

Step 1.2

Factor $\sin^{4} (x)$ out of $\sin^{6} (x)$ .

$\int \frac{\cos^{4} (x) \cdot 1}{\sin^{4} (x) \sin^{2} (x)} d x$

Step 1.3

Separate fractions.

$\int \frac{1}{\sin^{2} (x)} \cdot \frac{\cos^{4} (x)}{\sin^{4} (x)} d x$

Step 1.4

Convert from $\frac{\cos^{4} (x)}{\sin^{4} (x)}$ to $\cot^{4} (x)$ .

$\int \frac{1}{\sin^{2} (x)} \cot^{4} (x) d x$

Step 1.5

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

$\int \frac{1^{2}}{\sin^{2} (x)} \cot^{4} (x) d x$

Step 1.6

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

$\int {(\frac{1}{\sin (x)})}^{2} \cot^{4} (x) d x$

Step 1.7

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\int \csc^{2} (x) \cot^{4} (x) d x$

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 $d u$ .

$\int - 1 u^{4} d u$

Step 3

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

$- \int u^{4} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$- (\frac{1}{5} u^{5} + C)$

Step 5

Rewrite $- (\frac{1}{5} u^{5} + C)$ as $- \frac{1}{5} u^{5} + C$ .

$- \frac{1}{5} u^{5} + C$

Step 6

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

$- \frac{1}{5} \cot^{5} (x) + C$