   Chapter 7.3, Problem 14E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 1 d x ( x 2 + 1 ) 2

To determine

To evaluate: The given integral 01dx(x2+1)2.

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:

The integral, 01dx(x2+1)2

Calculation:

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

x=tanθdx=sec2θdθ

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

x0tanθ=0θ0andx1tanθ=1θπ4

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

01dx(x2+1)2=0π4sec2θdθ(tan2θ+1)2

Use the identity sec2x=1+tan2x:

01dx(x2+1)2=0

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