   Chapter 8.4, Problem 15E

Chapter
Section
Textbook Problem

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

To determine

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

Explanation

Given:

The provided integral is 1(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=2cos2θ1

Formula for trigonometric functions:

tanθ=perpendicularbase

sinθ=perpendicularhypotenuse

cosθ=basehypotenuse

Calculation:

Consider the integral 1(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=perpendicularhypotenuse=xx2+1

And

cosθ=basehypotenuse=1x2+1

And θ=arctanx. Substitute tanθ for x and sec2θdθ for dx in the integral 1(1+x2)2dx:

sec2θ(1+tan2θ)2dθ

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

Apply the trigonometric property and simplify:

1(1+x2)2dx=sec2θ(sec2θ)2dθ=sec2θsec4θdθ=1sec2θdθ

Evaluate:

1(1+x2)2dx=1sec

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