Calculus Examples

$\int \sin^{4} (θ) \cos (θ) d θ$ , $u = \sin (θ)$

Step 1

Let $u = \sin (θ)$ . Then $d u = \cos (θ) d x$ . Rewrite using $u$ and $d u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $\sin (θ)$ .

$\frac{d}{d θ} [\sin (θ)]$

Step 1.1.2

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

$\cos (θ)$

Step 1.2

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

$\int u^{4} d u$

Step 2

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

$\frac{1}{5} u^{5} + C$

Step 3

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

$\frac{1}{5} \sin^{5} (θ) + C$