   Chapter 7.3, Problem 23E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ d x x 2 + 2 x + 5

To determine

To evaluate: The given integral dxx2+2x+5.

Explanation

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

Formula used:

The identity, sec2x=1+tan2x

Given:

The integral, dxx2+2x+5

Calculation:

Rewrite the given integral by splitting the term 5 as 4+1:

dxx2+2x+5=dxx2+2x+4+1=dx(x+1)2+4

Use the substitution x+1=u, then du=dx. The integral in terms of u will be:

dxx2+2x+5=duu2+4

Substitute for u as u=2tanθ. Take the derivative of the substitution term:

u=2tanθdu=2sec2θdθ

Substitute for u and du in the given integral to get:

dxx2+2x+5=2sec2θdθ4tan2θ+4=sec2θdθtan2θ+1

Use the identity sec2x=1+tan2x:

dxx2

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