Chapter 5: Q. 14 (page 464)

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Expert verified

Ans:

$\sqrt{x^{2} + a^{2}} = a s e c u . . . . . . . [x = a \tan u] = \sqrt{a^{2} \tan^{2} u + a^{2}} = a s e c u$

Step by step solution

Step 1. Given information:

$x = a \tan u \sqrt{x^{2} + a^{2}} = a s e c u$

Step 2. Solving the trigonometric substitution:

If $x = a \tan u$ , there is no domain restriction on $x$ , because $x = a \tan u$ can be any real number. This means that trigonometric substitution $x = a \tan u x$ can be used even for integrands that do not have square root.

The domain of $x^{2} + a^{2}$ can be any value.

So, the integral involving $x^{2} + a^{2}$ are well suited for trigonometric substitution with $x = a \tan u$ .

Using the Pythagorean identity $a^{2} \tan^{2} u + a^{2} = a^{2} \sec^{2} u$ .

$\sqrt{x^{2} + a^{2}} = \sqrt{a^{2} \tan^{2} u + a^{2}} = \sqrt{a^{2} s e c^{2} u} = |a s e c u| [A . \sqrt{A^{2}} = |A|] = a s e c u$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Geometry

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Explain why, if x=atanu, then x2+a2=asecu. Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Geometry

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.