   Chapter 7.3, Problem 23E

Chapter
Section
Textbook Problem

# Evaluate the integral.23. ∫ d x x 2 + 2 x + 5

To determine

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

Explanation

Given information:

The integral function is dxx2+2x+5

Calculation:

Show the integral function as follows:

dxx2+2x+5 (1)

Rearrange Equation (1).

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

Consider x+1=2tanθ (3)

Differentiate both sides of the Equation (3).

dx=2sec2θdθ

Substitute (2tanθ) for (x+1) and (2sec2θdθ) for dx in Equation (2).

dx(x+1)2+4=(2sec2θdθ)(2tanθ)2+4=2sec2θ4tan2θ+4dθ=2sec2θ4(tan2θ+1)dθ=2sec2θ2(sec2θ)dθ

=sec2θsecθdθ=secθdθ (4)

Integrate Equation (4)

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