   Chapter 7.3, Problem 27E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ x 2 + 2 x   d x

To determine

To evaluate: The given integral x2+2xdx.

Explanation

Integration involving terms of the form x2a2 can be simplified by using the trigonometric substitution x=asecθ.

Formula used:

The identity, tan2x=sec2x1

and (ab)2=a22ab+b2

Given:

The integral, x2+2xdx

Calculation:

The integral can be written as:

x2+2xdx=x2+2x+11dx

Use the identity (a+b)2=a2+2ab+b2:

x2+2xdx=x2+2x+11dx=(x+1)21dx

Let u=x+1, then du=dx and the integration in terms of u will be:

x2+2xdx=u21dx

Now, substitute u=secθ,du=secθtanθdθ, then the integration becomes:

x2+2xdx=u21dx=sec2θ1secθtanθdθ

Here, 0θ<π2. Simplify further using the identity tan2x=sec2x1:

x2+2xdx=tan2θsecθtanθdθ=secθtan2θdθ=secθ(sec2θ1)dθ

=sec3θdθsecθdθ…… (1)

Solve each integral one by one. Consider the integral sec3θdθ, solve it by using integration by parts with u=secθ,dv=sec2θdθ, then du=secθtanθdθ,v=tanθ

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