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Found in: Page 326

### Essential Calculus: Early Transcendentals

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

# Evaluate the integral $$\int {{{\sin }^3}\theta {{\cos }^4}} \theta d\theta$$

Transform the integrand in the form $$\left( {{{\cos }^4}\theta - {{\cos }^6}\theta } \right)\sin \theta$$

First of all, we transform the given integrand in the form ‘$$\left( {{{\cos }^4}\theta - {{\cos }^6}\theta } \right)\sin \theta$$’, now we consider $$u = \cos \theta$$ followed by its differentiation with respect to $$\theta$$ then we substitute the value of ‘$$d\theta$$’ in the integral, on the integration of which we get a general solution. After putting back the value of u in the general solution we obtain the final answer.

See the step by step solution

## Step-1:Given data

Given integral is $$\int {{{\sin }^3}\theta {{\cos }^4}\theta d\theta }$$

\begin{aligned}{l} &= \int {{{\sin }^2}\theta {{\cos }^4}\theta } d\theta \\ &= \int {\left( {1 - {{\cos }^2}\theta } \right)} {\cos ^4}\theta \sin \theta d\theta \\ &= \int {\left( {{{\cos }^4}\theta - {{\cos }^6}\theta } \right)} \sin \theta d\theta \to (1)\end{aligned}

Now let $$u = \cos \theta$$ , differentiating ‘$$u$$’ with respect to ‘$$\theta$$’ we get,

$$\frac{{du}}{{d\theta }} = - \sin \theta ,d\theta = - \frac{{du}}{{\sin \theta }}$$

Now, putting the value of $$d\theta$$ in (1), we get the integral as

\begin{aligned}{l} &= \int {\left( {{u^4} - {u^6}} \right)} \sin \theta \left( { - \frac{1}{{\sin \theta }}} \right)du\\ &= \int { - {u^4}du + \int {{u^6}du} } \\ &= - \int {{u^4}du + \int {{u^6}du} } \end{aligned}

## Step-2:Integration

On integrating we get,

\begin{aligned}{l} - \int {{u^4}du + \int {{u^6}du} } \\\end{aligned}

$$= - \frac{1}{5}{u^5} + \frac{1}{7}{u^7} + c$$ , where ‘c’ is an arbitrary constant

Now on substituting the value of u, we get the solution as

$$- \frac{1}{5}{\cos ^5}\theta + \frac{1}{7}{\cos ^7}\theta + c$$

Hence, The general solution of the integral $$\int {{{\sin }^3}\theta {{\cos }^4}\theta d\theta }$$ is $$- \frac{1}{5}{\cos ^5}\theta + \frac{1}{7}{\cos ^7}\theta + c$$ , where ‘c’ is an arbitrary constant