How do you integrate 105x2−2x3 using partial fractions?

1 Answer
Narad T.
Feb 2, 2017

The answer is =−2x+45ln(|x|)−45ln(|2x−5|)+C

Explanation:

5x2−2x3=−x2(2x−5)

Let's perform the decompostion into partial fractions

105x2−2x3=−10(x2(2x−5))

=Ax2+Bx+C2x−5

=A(2x−5)+Bx(2x−5)+Cx2(x2(2x−5))

The denpminators are the same, we can compare the numerators

−10=A(2x−5)+Bx(2x−5)+Cx2

Let x=0, ⇒, −10=−5A, ⇒, A=2

Let x=52, ⇒, −10=254C, ⇒, C=−85

Coefficients of x2

0=2B+C, ⇒, B=−C2=12⋅85=45

Therefore,

105x2−2x3=2x2+45x+−852x−5

So,

∫10dx5x2−2x3=2∫dxx2+45∫dxx−85∫dx2x−5

=−2x+45ln(|x|)−45ln(|2x−5|)+C

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