   Chapter 8.4, Problem 16E

Chapter
Section
Textbook Problem

# Using Trigonometric Substitution In Exercises 13–16, find the indefinite integral using the substitution x = tan θ . ∫ x 2 ( 1 + x 2 ) 2 d x

To determine

To calculate: The value of indefinite integral x2(1+x2)2dx by the substitution of x=tanθ.

Explanation

Given:

The provided integral is x2(1+x2)2dx and substitute x=tanθ.

Formula used:

Power rule:

xndx=xn+1n+1+C

Where, x is a variable and n is a constant value.

Cosine rule:

cosθ=sinθ+C

Sine rule:

sin2θ=2sinθcosθ

Trigonometric identity:

cos2θ=12sin2θ

Calculation:

Consider the equation x2(1+x2)2dx. Put x=tanθ, now differentiate both sides with respect to x, dx=sec2θdθ

As

tanθ=perpendicularbase=x1

So, hypotenuse=1+x2.

sinθ=ph=xx2+1,cosθ=bh=1x2+1andθ=arctanx

Substitute tanθ for x and sec2θdθ for dx in the integral x2(1+x2)2dx: tan2θsec2θ(1+tan2θ)2dθ.

Recall the formula of trigonometric property, tan2θ+1=sec2θ

Apply the trigonometric property and simplify:

x2(1+x2)2dx=tan2θsec2θ(1+tan2θ)2dθ=tan2θsec2θ(sec2θ)2dθ=tan2θsec2θsec4θdθ=tan2θsec2θdθ

Evaluate:

x2(1+x2)2dx=tan2θsec2θdθ=&

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