   Chapter 7.3, Problem 7E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 a d x ( a 2 + x 2 ) 3 / 2    a > 0

To determine

To evaluate: The given integral 0adx(a2+x2)32,     a>0.

Explanation

Integration involving terms of the form a2+x2 can be simplified by using the trigonometric substitution x=atanθ.

Formula used:

The identity, sec2x=1+tan2x

Given:

Calculation:

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

x=atanθdx=asec2θdθ

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

x0atanθ=0θ0andxaatanθ=atanθ=1θπ4

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

Use the identity

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