Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Short Answer

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 by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

Find Study Materials for

Create Study Materials

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral \(\int {{{\sin }^3}\theta {{\cos }^4}} \theta d\theta \)

Step by Step Solution

Step-1:Given data

Step-2:Integration

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral \(\int {{{\sin }^3}\theta {{\cos }^4}} \theta d\theta \)

Step by Step Solution

Step-1:Given data

Step-2:Integration

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths