How do you integrate x+1(2x4)(x4)(x7) using partial fractions?

1 Answer
Dec 16, 2017

320ln(|x2|)512(|x4|)+415ln(|x7|)+C

Explanation:

x+1(2x4)(x4)(x7)

=12x+1(x2)(x4)(x7)

I decomposed x+1(x2)(x4)(x7) into basic fractions,

x+1(x2)(x4)(x7)

=Ax2+Bx4+Cx7

After expanding denominator,

A(x4)(x7)+B(x2)(x7)+C(x2)(x4)=x+1

Set x=2, 10A=3, so A=310

Set x=4, 6B=5, so B=56

Set x=7, 15C=8, so C=815

Thus,

x+1(2x4)(x4)(x7)dx

=12x+1(2x4)(x4)(x7)dx

=12310dxx21256dxx4+12815dxx7

=320dxx2512dxx4+415dxx7

=320ln(|x2|)512(|x4|)+415ln(|x7|)+C