Chapter 8.5, Problem 22E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Evaluating a Definite Integral In Exercises 21-24, use partial fractions to evaluate the definite integral. Use a graphing utility to verify your result. ∫ 1 5 x − 1 x 2 ( x + 1 ) d x .

To determine
The value of given definite integral by using partial fractions.

Explanation

Given:

The definite integral âˆ«15xâˆ’1x2(x+1)dx.

Formula used:

âˆ«1xdx=ln|x|+C.

Calculation:

Consider the following definite integral,

âˆ«15xâˆ’1x2(x+1)dx.

Now by using partial fraction method we get,

xâˆ’1x2(x+1)=Ax+Bx2+Cx+1MultiplyingÂ byÂ theÂ leastÂ commonÂ Â denominatorÂ x2(x+1)xâˆ’1x2(x+1)x2(x+1)=Axx2(x+1)+Bx2x2(x+1)+Cx+1x2(x+1)

Then,

xâˆ’1=Ax(x+1)+B(x+1)+Cx2xâˆ’1=Ax2+Ax+Bx+B+Cx2xâˆ’1=(A+C)x2+(A+B)x+B.

ByÂ equatingÂ theÂ CoefficientsÂ weÂ get,.

A+C=0â†’A=âˆ’CA+B=1â†’A=1âˆ’BB=âˆ’1A=1âˆ’Bâ†’A=1âˆ’(âˆ’1)â†’A=2C=âˆ’2

Substitute the values of A, B and C to get,

xâˆ’1x2(x+1)=2x+âˆ’1x2+âˆ’2x+1.........(1)

Now, solve the integral using relation in (1) to get,

âˆ«15xâˆ’1x2(x+1)dx=âˆ«15(2x+âˆ’1x2+âˆ’2x+1)dx

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