/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Evaluate the integral using (a) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 34

Evaluate the integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution. $$ \int_{0}^{\sqrt{3} / 2} \frac{1}{\left(1-t^{2}\right)^{5 / 2}} d t $$

Short Answer

Expert verified
After evaluation, the integral is equal to \( \frac{3}{8}\theta + \frac{1}{4}sin(2\theta) + \frac{1}{32}sin(4\theta) \) evaluated from 0 to \( \pi / 3 \) and comes out to be \( \frac{1}{32} \pi - \frac{1}{16} \).

Step by step solution

01

Identify Substitution

In the equation, we will start by identifying a suitable substitution. We can set \( t = sin\theta \).
02

Change the Limits

Substitute the old limits (0 to \( \sqrt{3} / 2 \)) with the new limits. These are obtained by substituting the old limits into the substitution identified in step 1. So we have \( 0 -> sin^{-1} 0 = 0 \) and \( \sqrt{3} / 2 -> sin^{-1} \sqrt{3}/2 = \pi / 3 \). Therefore, new limits of integration are 0 to \( \pi / 3 \).
03

Substitution in Integral

Now, substitute \( t = sin\theta \) into the original integral and transform \( dt \) to \( cos\theta d\theta \). It becomes \( \int \frac{cos\theta d\theta}{(1 - sin^{2}\theta)^{5 / 2}} \). Using Pythagorean identity it simplifies to \( \int cos^{4}\theta d\theta \)
04

Evaluate the Integral

Using the power-reducing identity, evaluate the integral. It becomes \( \int \frac{3}{8} + \frac{1}{2}cos(2\theta) + \frac{1}{8}cos(4\theta) d\theta \). The function can be integrated from 0 to \( \pi / 3 \) resulting in the final answer.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the integral. Use a computer algebra system to confirm your result. $$ \int\left(\tan ^{4} t-\sec ^{4} t\right) d t $$

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).

Consider the function \(h(x)=\frac{x+\sin x}{x}\). (a) Use a graphing utility to graph the function. Then use the zoom and trace features to investigate \(\lim _{x \rightarrow \infty} h(x)\). (b) Find \(\lim _{x \rightarrow \infty} h(x)\) analytically by writing \(h(x)=\frac{x}{x}+\frac{\sin x}{x}\) (c) Can you use L'H么pital's Rule to find \(\lim _{x \rightarrow \infty} h(x) ?\) Explain your reasoning.

Find the integral. Use a computer algebra system to confirm your result. $$ \int \tan ^{4} \frac{x}{2} \sec ^{4} \frac{x}{2} d x $$

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.