Chapter 8.4, Problem 45E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Comparing Methods(a) Find the integral ∫ x 1 − x 2 d x using u-substitution.Then find the integral using trigonometric substitution. Discuss the results.(b) Find the integral ∫ x 2 x 2 + 9 d x algebraically using x 2 = ( x 2 + 9 ) − 9 . Then find the integral using trigonometric substitution. Discuss the results.

(a)

To determine
Theintegral of following function by using the method of u-substitution and trigonometric substitution, x1x2dx.

Explanation

Consider following expression

âˆ«x1âˆ’x2dx â€¦â€¦â€¦(1)

Let 1âˆ’x2=u;

We know the derivative of the term 1âˆ’x2 is âˆ’2x.

âˆ’2x=u

Now, differentiate above expression with respect to x;

âˆ’2xdx=duxdx=du2

Substitute above values in equation (1) and integrate;

âˆ«x1âˆ’x2dx=âˆ«1u(âˆ’du2)âˆ«x1âˆ’x2dx=âˆ’12âˆ«uâˆ’12dxâˆ«x1âˆ’x2dx=âˆ’12u1212+Câˆ«x1âˆ’x2dx=âˆ’u+C

Substitute u=1âˆ’x2 to get;

âˆ«x1âˆ’x2dx=âˆ’1âˆ’x2

(b)

To determine
The solutions of the following indefinite integral x2x2+9dx algebraically using x2=(x2+9)9 and trigonometric substitution.

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