Q. 45 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 45 (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.
$鈭� \frac{\sqrt{9 - x^{2}}}{x} d x$

Short Answer

Expert verified

The solution of the integral is $3 \ln |\tan (\frac{1}{2} \sin^{- 1} \frac{x}{3})| + \sqrt{9 - x^{2}} + C .$

Step by step solution

Step 1. Given Information.

The given integral is $鈭� \frac{\sqrt{9 - x^{2}}}{x} d x .$

Step 2. Solve.

To solve the integral, let $x = 3 \sin u,$ so derivation of uis $d x = 3 \cos u d u .$

Thus, substitute u into the original integral,

role="math" localid="1648802752019" $鈭� \frac{\sqrt{9 - x^{2}}}{x} d x = 鈭� \frac{\sqrt{9 - {(3 \sin u)}^{2}}}{(3 \sin u)} 3 \cos u d u = 鈭� \frac{\sqrt{9 - 9 \sin^{2} u}}{\sin u} \cos u d u = 3 鈭� \frac{\sqrt{1 - \sin^{2} u}}{\sin u} \cos u d u Let' s use the identity \sin^{2} x + \cos^{2} x = 1 = 3 鈭� \frac{\sqrt{\cos^{2} u}}{\sin u} \cos u d u = 3 鈭� \frac{\cos^{2} u}{\sin u} d u Again, Let' s use the identity \sin^{2} x + \cos^{2} x = 1 = 3 鈭� \frac{1 - \sin^{2} u}{\sin u} d u = 3 鈭� \frac{1}{\sin u} d u - 3 鈭� \frac{\sin^{2} u}{\sin u} d u = 3 鈭� \cos e c u d u - 3 鈭� \sin u d u$

Step 3. Solve.

By proceeding with the calculation further,

$= 3 鈭� \cos e c u d u - 3 鈭� \sin u d u = 3 \ln |\tan (\frac{u}{2})| + 3 \cos u + C$

Now, substitute back uin the above equation,

$= 3 \ln |\tan (\frac{1}{2} \sin^{- 1} \frac{x}{3})| + 3 \cos (\sin^{- 1} \frac{x}{3}) + C = 3 \ln |\tan (\frac{1}{2} \sin^{- 1} \frac{x}{3})| + \sqrt{9 - x^{2}} + C$

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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.
$鈭� \frac{\sqrt{9 - x^{2}}}{x} d x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Solve.

Step 3. Solve.

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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�9-x2xdx

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

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Decision Maths

Geometry

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.
$鈭� \frac{\sqrt{9 - x^{2}}}{x} d x$