   Chapter 7.4, Problem 27E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ x 3 + 4 x + 3 x 4 + 5 x 2 + 4 d x

To determine

To evaluate the integral x3+4x+3x4+5x2+4dx

Explanation

Calculation: Given x3+4x+3x4+5x2+4dx

We can Factor the denominator as

x3+4x+3x4+5x2+4=x3+4x+3(x2+4)(x2+1)

Partial fraction decomposition is given by

x3+4x+3(x2+4)(x2+1)=Ax+B(x2+4)+Cx+D(x2+1)

x3+4x+3(x2+4)(x2+1)=(Ax+B)(x2+1)+(Cx+D)(x2+4)(x2+4)(x2+1)

x3+4x+3(x2+4)(x2+1)=(Ax3+Ax+Bx2+B)+(Cx3+4Cx+Dx2+4D)(x2+4)(x2+1)

x3+4x+3(x2+4)(x2+1)=Ax3+Ax+Bx2+Bx+Cx3+4Cx+Dx2+4D(x2+4)(x2+1)

x3+4x+3(x2+4)(x2+1)=(Ax3+Cx3)+(Bx2+Dx2)+(Ax+4Cx)+(B+4D)(x2+4)(x2+1)

x3+4x+3(x2+4)(x2+1)=(A+C)x3+(B+D)x2+(A+4C)x+(B+4D)(x2+4)(x2+1)

Set each coefficient on the left side of the equation equal to their respective coefficient on the right:

(1)x3+(0)x2+4x+3=(A+C)x3+(B+D)x2+(A+4C)x+(B+4D)

1=A+C<

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