How do you integrate ∫x+1(x−9)(x+8)(x−2) using partial fractions?

1 Answer
Narad T.
Jul 10, 2017

The answer is =10119ln(|x−9|)−7170ln(|x+8|)−370ln(|x−2|)+C

Explanation:

Let's perform the decomposition into partial fractions

x+1(x−9)(x+8)(x−2)=Ax−9+Bx+8+Cx−2

=A(x−2)(x+8)+B(x−9)(x−2)+C(x−9)(x+8)(x−9)(x+8)(x−2)

The denominator is the same, we compare the numerator.

x+1=A(x−2)(x+8)+B(x−9)(x−2)+C(x−9)(x+8)

Let x=9, ⇒, 10=17⋅7A, ⇒, A=10119

Let x=−8, ⇒, −7=−17⋅−10B, ⇒, B=−7170

Let x=2, ⇒, 3=−7⋅10C, ⇒, C=−370

Therefore,

x+1(x−9)(x+8)(x−2)=10119x−9−7170x+8−370x−2

∫(x+1)dx(x−9)(x+8)(x−2)=∫10119dxx−9−∫7170dxx+8−∫370dxx−2

=10119ln(|x−9|)−7170ln(|x+8|)−370ln(|x−2|)+C

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