   Chapter 7.4, Problem 58E

Chapter
Section
Textbook Problem

# Evaluate the integral by completing the square and using Formula 6. ∫ 2 x + 1 4 x 2 + 12 x − 7 d x

To determine

Evaluate the integral by completing the square and using Formula 6,2x+14x2+12x7dx

Explanation

Calculation: Given 2x+14x2+12x7dx

Since the derivative of the denominator can be written as numerator, we go ahead with the simplification

2x+14x2+12x7dx=148x+44x2+12x7dx=148x+1284x2+12x7dx

=148x+124x2+12x7dx+1484x2+12x7dx=148x+124x2+12x7dx214x2+12x7dx

The first integral can be directly written as

148x+124x2+12x7dx=14ln|4x2+12x7|+C

This is because numerator is derivative of denominator and the integral is of du/u form.

The second integral should be made a perfect square and then evaluated

214x2+12x7dx=214(x2+3x)7dx

=214(x+32)297dx=241(x+32)24dx=121(x+32)24dx

We Know that duu2a2=12aln|uau+a|+C

Let u=x+32,du=dx

Here a=2

121(x+32)24dx=12

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