   Chapter 8.4, Problem 45E

Chapter
Section
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

x1x2dx ………(1)

Use u-substitution as asked,

Let 1x2=u;

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

2x=u

Now, differentiate above expression with respect to x;

2xdx=duxdx=du2

Substitute above values in equation (1) and integrate;

x1x2dx=1u(du2)x1x2dx=12u12dxx1x2dx=12u1212+Cx1x2dx=u+C

Substitute u=1x2 to get;

x1x2dx=1x2

(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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