Advanced Integration Techniques

Consider $\int \frac{x+2}{x^2-1}dx$. Substitution fails (numerator isn't the derivative of the denominator) and by-parts goes nowhere. Instead, factor and split:

Solving $A(x+1) + B(x-1) = x + 2$ gives $A = \frac{3}{2}$, $B = -\frac{1}{2}$, so the integral is $\frac{3}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C$. This works whenever the denominator factors into distinct linear terms.