Chapter 7.3, Problem 32E

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

# Evaluate ∫ x 2 ( x 2 + a 2 ) 3 / 2   d x (a) by trigonometric substitution.(b) by the hyperbolic substitution x = a   sinh   t .

To determine

(a)

To evaluate: The given integral x2(x2+a2)32dx by trigonometric substitution.

Explanation

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

Formula used:

The derivative of xn is nxn1dx.

The identity tan2x+1=sec2x

Given:

The integral, x2(x2+a2)32dx

Calculation:

Use the substitution x=atanθ to simplify the integral. Then. The differential of x will be:

dx=asec2θdθ

The integration in terms of θ will be:

x2(x2+a2)32dx=a2tan2θ(a2tan2θ+a2)32asec2θdθ=a2tan2θa3(tan2θ+1)32asec2θdθ=tan2θ(tan2θ+1)32sec2θdθ

Use the identity tan2x+1=sec2x

x2(x2+a2)32dx=tan2θ(sec2θ)32sec2θdθ=tan2θsec3θsec2θdθ=tan2θsecθdθ

Rewrite the integral in terms of sine and cosine:

x2(x2+a2)32dx=tan2θ1secθdθ=sin2θcos2θcosθdθ=sin2θcosθdθ

Use the identity sin2x=1

To determine

(b)

To evaluate: The integral x2(x2+a2)32dx using hyperbolic substitution.

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