Distinct linear factors

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Place a constant over each distinct linear factor. Multiply by the common denominator and match coefficients or substitute roots to find the constants. Recombine the fractions to verify the identity.

Step-by-step example

1. Factor the denominator.

   $$\frac1{x^2-1}=\frac{A}{x-1}+\frac{B}{x+1}$$

2. Multiply and substitute x = 1 and x = −1.

   $$1=A(x+1)+B(x-1),\quad A=\tfrac12,\ B=-\tfrac12$$

3. Integrate both logarithms on intervals.

   $$I=\tfrac12\ln|x-1|-\tfrac12\ln|x+1|+C$$