Partial Fraction Decomposition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Integrate ∫x2+2x−1x(x−1)2 dx\displaystyle\int \frac{x^2+2x-1}{x(x-1)^2}\,dx.

Solution

  1. 1
    Ax+Bx−1+C(x−1)2\frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^2}.
  2. 2
    x=0x=0: A=−1A=-1. x=1x=1: C=2C=2. x2x^2 coeff: A+B=1⇒B=2A+B=1 \Rightarrow B=2.
  3. 3
    ∫=−ln⁡∣x∣+2ln⁡∣x−1∣−2x−1+C\int = -\ln|x|+2\ln|x-1|-\frac{2}{x-1}+C.

Answer

−ln⁡∣x∣+2ln⁡∣x−1∣−2x−1+C-\ln|x|+2\ln|x-1|-\frac{2}{x-1}+C
Repeated linear factors require extra terms: one per power up to the multiplicity.

About Partial Fraction Decomposition

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

Learn more about Partial Fraction Decomposition →

More Partial Fraction Decomposition Examples

Example 1 easy

Decompose [formula] and integrate.

Example 3 easy

Decompose [formula].

Example 4 medium

Integrate [formula].

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