   Chapter 8.4, Problem 13E

Chapter
Section
Textbook Problem

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

To determine

To calculate: The indefinite integral x1+x2dx by the use of substitution x=tanθ.

Explanation

Given:

The provided integral is x1+x2dx and substitute x=tanθ.

Formula used:

Power rule:

xndx=xn+1n+1+C

Trigonometric property:

tan2θ+1=sec2θ

Calculation:

Consider the integral x1+x2dx.

Take x=tanθ, then differentiate on both sides with respect to x to get dx=sec2θdθ.

The triangle is;

Substitute tanθ for x and sec2θdθ for dx in the integral x1+x2dx:

tanθ1+tan2θsec2θdθ

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

Apply the trigonometric property and simplify:

tanθsec3θdθ

Now let u=secθ and du=secθtanθdθ.

Evaluate the integral using the substitution:

x1+x2dx=tanθsec3θdθ=u2du

Recall the formula for power rule, xndx=xn+1n+1+C

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