   Chapter 7.4, Problem 31E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 1 x 3 − 1 d x

To determine

To evaluate the integral 1x31dx

Explanation

Calculation: Given 1x31dx

Resolve into partial fractions

1x31=1(x1)(x2+x+1)=1(x1)(x2+x+1)=Ax1+Bx+Cx2+x+11=A(x2+x+1)+(Bx+C)(x1)=(A+B)x2+(AB+C)x+(AC)on solving , we getA+B=0,AB+C=0,AC=1A=13,B=13,C=231x31=Ax1+Bx+Cx2+x+1=13(1x1)13(xx2+x+1)23(1x2+x+1)=13(1x1)13(x(x+12)2+34)23(1(x+12)2+34)=13(1x1)13(x(x+12)2+(32)2)23(1(x+12)2+(32)2)

Substitute u=x+12,du=dx

1x31dx=[13(1x1)<

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