Calculus Examples

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

Step 1

Let $u = 4 - \cos (x)$ . Then $d u = \sin (x) d x$ , so $\frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 - \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 - \cos (x)$ .

$\frac{d}{d x} [4 - \cos (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- \cos (x)]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \cos (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$0 - \frac{d}{d x} [\cos (x)]$

Step 1.1.3.2

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

$0 - - \sin (x)$

Step 1.1.3.3

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (x)$

Step 1.1.3.4

Multiply $\sin (x)$ by $1$ .

$0 + \sin (x)$

Step 1.1.4

Add $0$ and $\sin (x)$ .

$\sin (x)$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{2}$ .

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

Step 4.2.2

Move $2$ to the left of $u^{2}$ .

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

Step 5

Replace all occurrences of $u$ with $4 - \cos (x)$ .

$- \frac{1}{2 {(4 - \cos (x))}^{2}} + C$