   Chapter 7.3, Problem 18E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ d x [ ( a x ) 2 − b 2 ] 3 / 2

To determine

To evaluate: The given integral dx[(ax)2b2]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, dx[(ax)2b2]32

Calculation:

Rewrite the given integral by taking out a2 as the common factor in the denominator:

dx[(ax)2b2]32=dx[a2x2b2]32=dxa3[x2ba22]32

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

x=basecθdx=basecθtanθdθ

Here, 0θ<π2

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

dxa3[x2ba22]32=basecθtanθdθa3[b2a2sec2θba22]32=basecθtanθdθa3b3a3[sec2θ1]32

Use the identity tan2x=sec2x1:

dxa3[x2ba22]32=basecθtanθdθa3b3a3[sec2θ1]32=basecθtanθdθa3b3a3[tan2θ]32=1ab2secθtanθdθ[tan2θ]32

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