Calculus Examples

$\int \frac{\cos^{3} (x)}{\sin^{4} (x)} d x$

Step 1

Simplify.

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Step 1.1

Multiply by $1$ .

$\int \frac{\cos^{3} (x) \cdot 1}{\sin^{4} (x)} d x$

Step 1.2

Factor $\sin^{3} (x)$ out of $\sin^{4} (x)$ .

$\int \frac{\cos^{3} (x) \cdot 1}{\sin^{3} (x) \sin (x)} d x$

Step 1.3

Separate fractions.

$\int \frac{1}{\sin (x)} \cdot \frac{\cos^{3} (x)}{\sin^{3} (x)} d x$

Step 1.4

Convert from $\frac{\cos^{3} (x)}{\sin^{3} (x)}$ to $\cot^{3} (x)$ .

$\int \frac{1}{\sin (x)} \cot^{3} (x) d x$

Step 1.5

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

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

Step 2

Write $\cot^{3} (x)$ as $\cot (x) \cot^{2} (x)$ .

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

Step 3

Using the Pythagorean Identity, rewrite $\cot^{2} (x)$ as $- 1 + \csc^{2} (x)$ .

$\int \csc (x) \cot (x) (- 1 + \csc^{2} (x)) d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\csc (x)$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 5

Multiply $- (- 1 + u^{2})$ .

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

Step 6

Multiply $- 1$ by $- 1$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

$u + C + \int - u^{2} d u$

Step 9

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

$u + C - \int u^{2} d u$

Step 10

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$u + C - (\frac{1}{3} u^{3} + C)$

Step 11

Simplify.

$u - \frac{1}{3} u^{3} + C$

Step 12

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

$\csc (x) - \frac{1}{3} \csc^{3} (x) + C$