Calculus Examples

$\int {(1 + \sin (x))}^{\frac{5}{2}} \cos (x) d x$

Step 1

Let $u = 1 + \sin (x)$ . 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 1.1

Let $u = 1 + \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + \sin (x)$ .

$\frac{d}{d x} [1 + \sin (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

$0 + \cos (x)$

Step 1.1.4

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

$\cos (x)$

Step 1.2

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

$\int u^{\frac{5}{2}} d u$

Step 2

By the Power Rule, the integral of $u^{\frac{5}{2}}$ with respect to $u$ is $\frac{2}{7} u^{\frac{7}{2}}$ .

$\frac{2}{7} u^{\frac{7}{2}} + C$

Step 3

Replace all occurrences of $u$ with $1 + \sin (x)$ .

$\frac{2}{7} {(1 + \sin (x))}^{\frac{7}{2}} + C$