/*! 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 26 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 26

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{1-2 x^{2}}}$$

Short Answer

Expert verified
Question: Evaluate the integral $\int \frac{d x}{\sqrt{1-2 x^{2}}}$ using the technique of trigonometric substitution. Answer: The evaluated integral is: $$\int \frac{d x}{\sqrt{1-2 x^{2}}} = \frac{1}{\sqrt{2}} \arcsin(\sqrt{2} x) + C$$.

Step by step solution

01

Choose a Trigonometric Substitution

In this case, since we have a square root function with a quadratic expression inside it, we'll use the substitution \(x = \frac{\sin\theta}{\sqrt{2}}\), so that the expression inside the square root becomes a trigonometric one. Differentiate both sides to find \(dx\): $$d x = \frac{\cos{\theta}}{\sqrt{2}} d\theta \tag{1}$$
02

Substitute variables into the Integral

Replace \(x\) by the trigonometric expression in the integral and replace \(dx\) by the expression found in \((1)\): $$\int \frac{\frac{\cos{\theta}}{\sqrt{2}} d\theta}{\sqrt{1-2 \left(\frac{\sin\theta}{\sqrt{2}}\right)^{2}}}$$
03

Simplify the Integral

Now, simplify the expression inside the square root and the integrand: $$\int \frac{\frac{\cos{\theta}}{\sqrt{2}} d\theta}{\sqrt{1 - \sin^{2}{\theta}}}$$ As we know, \(1 - \sin^{2}{\theta} = \cos^{2}{\theta}\), so we can further simplify the integral to: $$\int \frac{\frac{\cos{\theta}}{\sqrt{2}} d\theta}{\cos{\theta}}$$
04

Evaluate the Simplified Integral

Now that the integral is simplified, it's easy to evaluate: $$\int \frac{\frac{\cos{\theta}}{\sqrt{2}} d\theta}{\cos{\theta}} = \int \frac{1}{\sqrt{2}} d\theta$$ $$\frac{1}{\sqrt{2}} \int d\theta = \frac{1}{\sqrt{2}} \theta + C$$
05

Apply the Inverse Substitution

To find the integral in terms of \(x\), we need to apply the inverse substitution. First, we need to find \(\theta\) in terms of \(x\). From the original substitution \(x = \frac{\sin\theta}{\sqrt{2}}\), we can get the inverse substitution: $$\theta = \arcsin(\sqrt{2} x)$$ Now, substitute this expression back into the integral: $$\frac{1}{\sqrt{2}} \left(\arcsin(\sqrt{2} x)\right) + C$$
06

Final Answer

The evaluated integral is: $$\int \frac{d x}{\sqrt{1-2 x^{2}}} = \frac{1}{\sqrt{2}} \arcsin(\sqrt{2} x) + C$$

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

Water is drained from a 3000 -gal tank at a rate that starts at 100 gal/hr and decreases continuously by \(5 \% / \mathrm{hr}\). If the drain is left open indefinitely, how much water is drained from the tank? Can a full tank be emptied at this rate?

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{3 x^{2}+4 x-6}{x^{2}-3 x+2} d x$$

Find the volume of the following solids. The region bounded by \(y=1 /(x+2), y=0, x=0,\) and \(x=3\) is revolved about the line \(x=-1\)

$$\text { Evaluate } \int \frac{d x}{1+\sin x+\cos x} \text { using the }$$ substitution \(x=2 \tan ^{-1} \theta .\) The identities \(\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\) are helpful.

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{\sec \theta}{1+\sin \theta} d \theta$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.