Chapter 4: Q. 26 (page 404)

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.
$\int \frac{1}{1 - 2 x^{2}} d x$

Short Answer

Expert verified

The solution is: $- \frac{1}{2 \sqrt{2}} [\ln |\sqrt{2} x + 1| - \ln |\sqrt{2} x - 1|] + C$

Step by step solution

Given

The given integral is: $\int \frac{1}{1 - 2 x^{2}} d x$

To Find

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral.

Calculation

Let,

$u = \sqrt{2} x d u = \sqrt{2} d x \frac{d u}{\sqrt{2}} = d x$

Using substitution we get:

role="math" localid="1648935703243" $\int \frac{1}{1 - 2 x^{2}} d x = \int \frac{1}{1 - {(\sqrt{2} x)}^{2}} d x = \int \frac{1}{1 - u^{2}} \frac{d u}{\sqrt{2}} = \frac{1}{\sqrt{2}} \int \frac{1}{1 - u^{2}} d u = - \frac{1}{\sqrt{2}} \int \frac{1}{u^{2} - 1} d u = - \frac{1}{\sqrt{2}} \int \frac{1}{(u + 1) (u - 1)} d u = - \frac{1}{2 \sqrt{2}} \int \frac{2}{(u + 1) (u - 1)} d u = - \frac{1}{2 \sqrt{2}} \int \frac{(u + 1) - (u - 1)}{(u + 1) (u - 1)} d u = - \frac{1}{2 \sqrt{2}} \int [\frac{1}{u - 1} - \frac{1}{u + 1}] d u = - \frac{1}{2 \sqrt{2}} [\ln |u + 1| - \ln |u - 1|] + C = - \frac{1}{2 \sqrt{2}} [\ln |\sqrt{2} x + 1| - \ln |\sqrt{2} x - 1|] + C$

Where, C = integrating constant.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.
$\int \frac{1}{1 - 2 x^{2}} d x$

Short Answer

Step by step solution

Given

To Find

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Logic and Functions

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.∫11-2x2dx

Short Answer

Step by step solution

Given

To Find

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Logic and Functions

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.
$\int \frac{1}{1 - 2 x^{2}} d x$