Challenge Show that with the change of variables , the

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567 2

Solution for problem 82AE Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Problem 82AE

Challenge Show that with the change of variables , the integral can be converted to an integral amenable to partial fractions. Evaluate .

Step-by-Step Solution:

Problem 82AE

Challenge Show that with the change of variables $u=\sqrt{tan\ x}$ , the integral $\int\limits_{\ }^{\ }\sqrt{tan\ x}\ dx$ can be converted to an integral amenable to partial fractions. Evaluate $\int\limits_{0}^{\pi{}/4}\sqrt{tan\ x}\ dx$ .

Solution

Step 1

In this problem we have to check whether with the change of variables $u=\sqrt{tan\ x}$ , the integral $\int\limits_{\ }^{\ }\sqrt{tan\ x}\ dx$ can be converted to an integral amenable to partial fractions or not. And then we have to evaluate $\int\limits_{0}^{\pi{}/4}\sqrt{tan\ x}\ dx$

Consider $\int\limits_{\ }^{\ }\sqrt{tan\ x}\ dx$

Take $u=\sqrt{tan\ x}$

$\Rightarrow{}{u}^{2}=tan\ x$

$\Rightarrow{}x=arctan\ ({u}^{2})$

Differentiating we get $dx\ =\frac{1}{1\ +\ {u}^{4}}(2u)\ du\ =\frac{2u}{1\ +\ {u}^{4}}\ du$

Thus $\int\limits_{\ }^{\ }\sqrt{tan\ x}\ dx=\int\limits_{\ }^{\ }\frac{2{u}^{2}}{1\ +\ {u}^{4}}\ du$ which is an integral amenable to partial fractions.

Hence we have showed that with the change of variables $u=\sqrt{tan\ x}$ , the integral $\int\limits_{\ }^{\ }\sqrt{tan\ x}\ dx$ can be converted to an integral amenable to partial fractions.

Step 2 of 5

Chapter 7.4, Problem 82AE is Solved

Step 3 of 5

Textbook: Calculus: Early Transcendentals

Edition: 1

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567

The answer to “Challenge Show that with the change of variables , the integral can be converted to an integral amenable to partial fractions. Evaluate .” is broken down into a number of easy to follow steps, and 23 words. Since the solution to 82AE from 7.4 chapter was answered, more than 248 students have viewed the full step-by-step answer. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. The full step-by-step solution to problem: 82AE from chapter: 7.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: integral, fractions, change, converted, evaluate. This expansive textbook survival guide covers 112 chapters, and 5248 solutions.