Calculus Examples

$\frac{\cos (x)}{\sin^{5} (x)}$

Step 1

Write $\frac{\cos (x)}{\sin^{5} (x)}$ as a function.

$f (x) = \frac{\cos (x)}{\sin^{5} (x)}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{\cos (x)}{\sin^{5} (x)} d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\sin (x)$ .

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

Step 4.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 4.2

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

$\int \frac{1}{u^{5}} d u$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $u^{5}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{5})}^{- 1} d u$

Step 5.2

Multiply the exponents in ${(u^{5})}^{- 1}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int u^{5 \cdot - 1} d u$

Step 5.2.2

Multiply $5$ by $- 1$ .

$\int u^{- 5} d u$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $- \frac{1}{4} u^{- 4} + C$ as $- \frac{1}{4} \cdot \frac{1}{u^{4}} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{1}{u^{4}}$ by $\frac{1}{4}$ .

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

Step 7.2.2

Move $4$ to the left of $u^{4}$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Multiply by $1$ .

$- \frac{1}{4 (\sin^{4} (x) \cdot 1)} + C$

Step 9.2

Separate fractions.

$- (\frac{1}{4 \cdot (1)} \cdot \frac{1}{\sin^{4} (x)}) + C$

Step 9.3

Convert from $\frac{1}{\sin^{4} (x)}$ to $\csc^{4} (x)$ .

$- (\frac{1}{4 \cdot (1)} \csc^{4} (x)) + C$

Step 9.4

Multiply $4$ by $1$ .

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

Step 9.5

Combine $\frac{1}{4}$ and $\csc^{4} (x)$ .

$- \frac{\csc^{4} (x)}{4} + C$

Step 9.6

Reorder terms.

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

Step 10

The answer is the antiderivative of the function $f (x) = \frac{\cos (x)}{\sin^{5} (x)}$ .

$F (x) =$ $- \frac{1}{4} \csc^{4} (x) + C$