How do you integrate ∫x−3x2(x−6)(x−2)(x−5) using partial fractions?

1 Answer
Shwetank Mauria
Jul 4, 2016

∫x−3x2(x−6)(x−2)(x−5)dx

= −1218ln(x−6)+58ln(x−2)+354ln(x−5)+c

Explanation:

Let us first convert x−3x2(x−6)(x−2)(x−5) into partial fractions.

x−3x2(x−6)(x−2)(x−5)⇔Ax−6+Bx−2+Cx−5

x−3x2(x−6)(x−2)(x−5)⇔A(x−2)(x−5)+B(x−6)(x−5)+C(x−6)(x−2)(x−6)(x−2)(x−5)

= A(x2−7x+10)+B(x2−11x+30)+C(x2−8x+12)(x−6)(x−2)(x−5)

= x2(A+B+C)−x(7A+11B+8C)+(10A+30B+12C)(x−6)(x−2)(x−5)

Hence A+B+C=−3, 7A+11B+8C=−1 and 10A+30B+12C=0

Subtracting 7 times of first equation from second and 10 times first from third equation, we get

4B+C=20 and 20B+C=30 which gives us B=58 and C=1408=354 and putting these in A+B+C=−3, we get A=−1218

Hence, x−3x2(x−6)(x−2)(x−5)=−1218(x−6)+58(x−2)+354(x−5)

and ∫x−3x2(x−6)(x−2)(x−5)dx=∫[−1218(x−6)+58(x−2)+354(x−5)]dx

= −1218ln(x−6)+58ln(x−2)+354ln(x−5)+c

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