Q. 39 Solve each of the integrals in E... [FREE SOLUTION]

91影视

Chapter 5: Q. 39 (page 465)

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

Short Answer

Expert verified

The solution of the integral is $- \frac{\sqrt{4 - x^{2}}}{x} - \sin^{- 1} \frac{x}{2} + C .$

Step by step solution

Step 1. Given Information.

The given integral is $鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x .$

Step 2. Solve.

To solve the integral, let $x = 2 \sin u,$ so the derivation of u is $d x = 2 \cos u d u .$

Thus, substitute u into the original integral.

localid="1648797152038" $鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x = 鈭� \frac{\sqrt{4 - {(2 \sin u)}^{2}}}{{(2 \sin u)}^{2}} 2 \cos u d u = 鈭� \frac{\sqrt{4 - (4 \sin^{2} u)}}{4 \sin^{2} u} 2 \cos u d u = 鈭� \frac{\sqrt{4 (1 - \sin^{2} u)}}{4 \sin^{2} u} 2 \cos u d u = 鈭� \frac{2 \sqrt{1 - \sin^{2} u}}{4 \sin^{2} u} 2 \cos u d u = 鈭� \frac{\sqrt{1 - \sin^{2} u}}{\sin^{2} u} \cos u d u Let' s use the identity s i n^{2} x + c o s^{2} x = 1 = 鈭� \frac{\sqrt{\cos^{2} u}}{\sin^{2} u} \cos u d u = 鈭� \frac{\cos^{2} u}{\sin^{2} u} d u$

Step 3. Solve.

By proceeding with the calculation further,

$= 鈭� \frac{\cos^{2} u}{\sin^{2} u} d u = 鈭� c o t^{2} u d u = 鈭� c s c^{2} u - 1 d u = 鈭� c s c^{2} u d u - 鈭� 1 d u = - c o t u - u + C$

Now, substitute back uin the above equation,

$= - c o t (\sin^{- 1} (\frac{x}{2})) - \sin^{- 1} \frac{x}{2} + C Use the identity c o t (\sin^{- 1} (x)) = \frac{\sqrt{1 - x^{2}}}{x} = - \frac{\sqrt{1 - {(\frac{x}{2})}^{2}}}{\frac{x}{2}} - \sin^{- 1} \frac{x}{2} + C = - \frac{\sqrt{4 - x^{2}}}{x} - \sin^{- 1} \frac{x}{2} + C$

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

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

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Solve.

Step 3. Solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.鈭�4-x2x2dx

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Solve.

Step 3. Solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$