Chapter 7.3, Problem 23E

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636

Chapter
Section

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636
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 âˆ«dxx2+2x+5

Calculation:

Show the integral function as follows:

âˆ«dxx2+2x+5 (1)

Rearrange Equation (1).

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

Consider x+1=2tanÎ¸ (3)

Differentiate both sides of the Equation (3).

dx=2sec2Î¸â€‰dÎ¸

Substitute (2tanÎ¸) for (x+1) and (2sec2Î¸â€‰dÎ¸) for dx in Equation (2).

âˆ«dx(x+1)2+4=âˆ«(2sec2Î¸â€‰dÎ¸)(2tanÎ¸)2+4=âˆ«2sec2Î¸4tan2Î¸+4dÎ¸=âˆ«2sec2Î¸4(tan2Î¸+1)dÎ¸=âˆ«2sec2Î¸2(sec2Î¸)dÎ¸

=âˆ«sec2Î¸secÎ¸dÎ¸=âˆ«secÎ¸dÎ¸ (4)

Integrate Equation (4)

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