   Chapter 7.3, Problem 9E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 2 3 d x ( x 2 − 1 ) 3 / 2

To determine

To evaluate: The given integral 23dx(x21)32.

Explanation

Integration involving terms of the form x2a2 can be simplified by using the trigonometric substitution x=asecθ.

Formula used:

The identity, tan2x=sec2x1

Given:

The integral, 23dx(x21)32

Calculation:

Substitute for x as x=secθ. Take the derivative of the substitution term:

x=secθdx=secθtanθdθ

Here, 0θ<π2. The limits of integration will change as:

x2secθ=2θπ3andx3secθ=3θ=cos1(13)

Substitute for x and dx in the given integral to get:

23dx(x21)32=π3cos1(13)secθtanθdθ(sec2θ1)32

Use the identity tan2x=sec2x1:

23dx(x21)32=π3cos1(13)secθtanθdθ(tan2θ)32=π3cos1(13)secθtanθdθtan3θ=π3cos1(13)secθdθtan2θ

Write the integral in terms of sine and cosine:

23dx(x21)32=π3cos1(13)1cosθdθsin2θcos2θ=π3cos1(13)cosθsin2θdθ

Substitute u=sinθ,du=cosθdθ

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