How do you integrate ∫x−13x2−14x+15 using partial fractions?

1 Answer
Narad T.
Dec 4, 2016

The answer is =−16ln(∣3x−5∣)+12ln(∣x−3∣)+C

Explanation:

Let's factorise the denominator

3x2−14x+15=(3x−5)(x−3)

So, the decomposition into partial fractions are

x−13x2−14x+15=x−1(3x−5)(x−3)=A3x−5+Bx−3

=A(x−3)+B(3x−5)(3x−5)(x−3)

So, (x−1)=A(x−3)+B(3x−5)

Let x=3, ⇒, 2=4B, ⇒, B=12

Let x=53, ⇒, 23=−4A3, ⇒,A=−12

so,

x−13x2−14x+15=−123x−5+12x−3

∫(x−1)dx3x2−14x+15=∫(−12)dx3x−5+∫(12)dxx−3

=−12⋅ln(∣3x−5∣)3+12⋅ln(∣x−3∣)+C

=−16⋅ln(∣3x−5∣)+12⋅ln(∣x−3∣)+C

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