Calculus Examples

$\frac{\cos (x)}{{(2 + \sin (x))}^{2}}$

Step 1

Write $\frac{\cos (x)}{{(2 + \sin (x))}^{2}}$ as a function.

$f (x) = \frac{\cos (x)}{{(2 + \sin (x))}^{2}}$

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.

$F (x) = \int \frac{\cos (x)}{{(2 + \sin (x))}^{2}} d x$

Step 4

Let $u = 2 + \sin (x)$ . 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 $u = 2 + \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $2 + \sin (x)$ .

$\frac{d}{d x} [2 + \sin (x)]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $2 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [\sin (x)]$

Step 4.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 4.1.3

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

$0 + \cos (x)$

Step 4.1.4

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

$\cos (x)$

Step 4.2

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

$- u^{- 1} + C$

Step 7

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

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

Step 8

Replace all occurrences of $u$ with $2 + \sin (x)$ .

$- \frac{1}{2 + \sin (x)} + C$

Step 9

The answer is the antiderivative of the function $f (x) = \frac{\cos (x)}{{(2 + \sin (x))}^{2}}$ .

$F (x) =$ $- \frac{1}{2 + \sin (x)} + C$