   Chapter 7.R, Problem 25E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 3 x 3 − x 2 + 6 x − 4 ( x 2 + 1 ) ( x 2 + 2 ) d x

To determine

To Find:3x3x2+6x4(x2+1)(x2+2)dx

Explanation

Formula Used:dxx2+a2=1atan1(xa)+C

Calculation:

Here, the integrand is the ratio of two polynomials in which the degree of the numerator is less than the degree of the denominator. So, write the integrand in integrable form by using partial fraction as shown below:

3x3x2+6x4(x2+1)(x2+2)=Ax+Bx2+1+Cx+Dx2+23x3x2+6x4=(Ax+B)(x2+2)+(Cx+D)(x2+1)3x3x2+6x4=(A+C)x3+(B+D)x2+(2A+C)x+2B+D

Now, compare the relevant term and constant term to have

A+C=3, B+D=1, 2A+C=6, 2B+D=4

Solve these set of equations to find A, B, C and D.

A=3,B=3,C=0,D=2

The integral after rewriting the integrand is

3x3x2+6x4(x2+1)(x2+2)dx=(3x3x2

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