   Chapter 8.5, Problem 25E

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 2 x + 1 x ( x 2 + 1 ) d x .

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

Explanation

Given: The definite integral is 12x+1x(x2+1)dx.

Formula used:

1. 1xdx=ln|x|+C

2. dxx2+a2=1aarctanxa+C

Calculation:

Consider the following definite integral,

12x+1x(x2+1)dx.

Now by using partial fraction method we get,

x+1x(x2+1)=Ax+Bx+Cx2+1x+1x(x2+1)=A(x2+1)+(Bx+C)xx(x2+1)

By simplifying further, we get,

x+1=Ax2+A+Bx2+Cx x+1=(A+B)x2+Cx+A.

By equating the Coefficients we get,A+B=0C=1A=1A+B=0B=AB=1

Substituting the value of A, B and C to get,

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

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

12x+1x(x2+1)dx=12(1x+x+1x2+1)dx=12(1xxx2+1+1x2+1)dx

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