   Chapter 7.4, Problem 35E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 5 x 4 + 7 x 2 + x + 2 x ( x 2 + 1 ) 2 d x

To determine

To evaluate the integral 5x4+7x2+x+2x(x2+1)2dx

Explanation

Calculation: Given 5x4+7x2+x+2x(x2+1)2dx

Resolving into partial fractions

5x4+7x2+x+2x(x2+1)2=Ax+Bx+Cx2+1+Dx+E(x2+1)25x4+7x2+x+2x(x2+1)2=A(x2+1)2+x(Bx+C)(x2+1)+x(Dx+E)x(x2+1)2

Multiply both sides by the common denominator

5x4+7x2+x+2=A(x2+1)2+x(Bx+C)(x2+1)+x(Dx+E)5x4+7x2+x+2=A(x4+2x2+1)+(Bx2+Cx)(x2+1)+Dx2+Ex5x4+7x2+x+2=Ax4+2Ax2+A+Bx4+Bx2+Cx3+Cx+Dx2+Ex5x4+7x2+x+2=(Ax4+Bx4)+Cx3+(2Ax2+Bx2+Dx2)+(Cx+Ex)+A5x4+7x2+x+2=(A+B)x4+(C)x3+(2A+B+D)x2+(C+E)x+(A)

On comparing the coefficients of x4,x3,x2,x,x0

A=2,B=3,C=0,D=0,E=1

5x4+7x2+x+2x(x2+1)2=Ax+Bx+Cx2+1+Dx+E(x2+1)2=2x+(3)x+0x2+1+(0)x+1(x2+1)2=2x+3xx2+1+1(x2+1)2

I

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