How do you integrate int (x-9)/((x+8)(x-7)(x-5)) ∫x−9(x+8)(x−7)(x−5) using partial fractions?

1 Answer
bp
Feb 13, 2016

-17/195 ln(x+8) -1/15 ln(x-7)+2/13 ln (x-5)+ C_1−17195ln(x+8)−115ln(x−7)+213ln(x−5)+C1

Explanation:

To Integrate, partial fractions are required, which can be done as follows: Let,
(x-9)/((x+8)(x-7)(x-5))= A/(x+8) +B/(x-7) +C/(x-5) x−9(x+8)(x−7)(x−5)=Ax+8+Bx−7+Cx−5

=(A(x-7)(x-5) +B(x+8)(x-5) +C(x+8)(x-7))/((x+8)(x-7)(x-5))A(x−7)(x−5)+B(x+8)(x−5)+C(x+8)(x−7)(x+8)(x−7)(x−5)

Thus x-9= A(x^2-12x+35)+B(x^2 +3x-40)+C(x^2 +x-56)x−9=A(x2−12x+35)+B(x2+3x−40)+C(x2+x−56)

Comparing like coefficients, it would be
A+B+C=0, -12A +3B +C=1 and 35A -40B-56C=-9. Solving for A,B and C, it would be

A= -17/195, B= -1/15 and C= 2/13A=−17195,B=−115andC=213

The given integral would thus be -17/195 ln(x+8) -1/15 ln(x-7)+2/13 ln (x-5)+ C_1−17195ln(x+8)−115ln(x−7)+213ln(x−5)+C1

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