How do you find int x/((x+2)(x+9))dx using partial fractions?

1 Answer
Nov 3, 2015

9/7ln(x+9) - 2/7ln(x+2)+constant

Explanation:

The equation to be integrated can be written as:

x/((x+2)(x+9))=A/(x+2)+B/(x+9)=(A(x+9)+B(x+2))/((x+2)(x+9)

Equating coefficients

x->A(x+9)+B(x+2)=(A+B)x+(9A+2B)

x=(A+B)x

Therefore, dividing by x:

  • A+B=1 (1)

Additionally,

  • 9A+2B=0 (2)

Simultaneous Equations

(2)-2times(1)

9A+2B=0 (2)
-2A-2B=-2 (1)

Cancel out the B terms:

7A=-2

A=-2/7

Subbing A back into (1):

B=9/7

Integrating

Subbing A and B back into the original equation:

int9/(7(x+9))-2/(7(x+2))dx

=int9/(7(x+9))dx-int2/(7(x+2))dx

=9/7int1/(x+9)dx-2/7int1/(x+2)dx

=9/7ln(x+9) +constant -2/7ln(x+2)+constant

=9/7ln(x+9) - 2/7ln(x+2)+constant