   Chapter 8.5, Problem 22E

Chapter
Section
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 15x1x2(x+1)dx.

Formula used:

1xdx=ln|x|+C.

Calculation:

Consider the following definite integral,

15x1x2(x+1)dx.

Now by using partial fraction method we get,

x1x2(x+1)=Ax+Bx2+Cx+1Multiplying by the least common  denominator x2(x+1)x1x2(x+1)x2(x+1)=Axx2(x+1)+Bx2x2(x+1)+Cx+1x2(x+1)

Then,

x1=Ax(x+1)+B(x+1)+Cx2x1=Ax2+Ax+Bx+B+Cx2x1=(A+C)x2+(A+B)x+B.

By equating the Coefficients we get,.

A+C=0A=CA+B=1A=1BB=1A=1BA=1(1)A=2C=2

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

x1x2(x+1)=2x+1x2+2x+1.........(1)

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

15x1x2(x+1)dx=15(2x+1x2+2x+1)dx

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