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Found in: Page 306

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# Evaluate the integral by making the given substitution.$$\int {{{\cos }^3}} \theta \sin \theta d\theta ,\;\;\;u = \cos \theta$$

The value of $$\int {{{\cos }^3}} \theta \sin \theta d\theta$$is$$- \frac{{{{(\cos \theta )}^4}}}{4} + C$$.

See the step by step solution

## Step 1 Definition of the integral

Integral calculus is an area of mathematics that deals with the calculation, properties, and applications of integrals.

## Step 2 : Evaluating the integral.

The given data is expressed as,

$$\int {{{\cos }^3}} \theta \sin \theta d\theta$$

After Substituting

$$\cos \theta = u$$

and

\begin{aligned}{c} - \sin \theta d\theta &= du\\\sin \theta d\theta &= - du\end{aligned}

\begin{aligned}{c}\int {{{\cos }^3}} \theta \sin \theta d\theta &= - \int {{u^3}} du\\ &= - \frac{{{u^4}}}{4} + C\end{aligned}

Substituting again

$$u = \cos \theta$$,

We get

$$\int {{{\cos }^3}} \theta \sin \theta d\theta = - \frac{{{{(\cos \theta )}^4}}}{4} + C$$

Hence the value of $$\int {{{\cos }^3}} \theta \sin \theta d\theta$$is$$- \frac{{{{(\cos \theta )}^4}}}{4} + C$$.