Calculus Examples

$\int \sin (x) \sqrt{2 + 3 \cos (x)} d x$

Step 1

Let $u = 2 + 3 \cos (x)$ . Then $d u = - 3 \sin (x) d x$ , so $- \frac{1}{3} d u = \sin (x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 + 3 \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 + 3 \cos (x)$ .

$\frac{d}{d x} [2 + 3 \cos (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $2 + 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [3 \cos (x)]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [3 \cos (x)]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [3 \cos (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x)]$ .

$0 + 3 \frac{d}{d x} [\cos (x)]$

Step 1.1.3.2

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

$0 + 3 (- \sin (x))$

Step 1.1.3.3

Multiply $- 1$ by $3$ .

$0 - 3 \sin (x)$

Step 1.1.4

Subtract $3 \sin (x)$ from $0$ .

$- 3 \sin (x)$

Step 1.2

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

$\int \sqrt{u} \frac{1}{- 3} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$\int - \frac{\sqrt{u}}{3} d u$

Step 3

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

$- \int \frac{\sqrt{u}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$- \frac{1}{3} \int u^{\frac{1}{2}} d u$

Step 6

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

$- \frac{1}{3} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $- \frac{1}{3} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $- \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$- \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{2}{3}$ by $\frac{1}{3}$ .

$- \frac{2}{3 \cdot 3} u^{\frac{3}{2}} + C$

Step 7.2.2

Multiply $3$ by $3$ .

$- \frac{2}{9} u^{\frac{3}{2}} + C$

Step 8

Replace all occurrences of $u$ with $2 + 3 \cos (x)$ .

$- \frac{2}{9} {(2 + 3 \cos (x))}^{\frac{3}{2}} + C$