Integration Using Partial Fractions

Splitting a rational function into simpler pieces

When the integrand is a rational function $\frac{p(x)}{q(x)}$ with $q(x)$ factoring into distinct linear factors, the partial fraction decomposition rewrites it as a sum of simpler fractions that are each directly integrable using $\int\frac{1}{x-a}\,dx=\ln|x-a|+C$.

For two distinct linear factors:

where $A,B$ are found by clearing denominators and comparing coefficients (or by substituting $x=a$ and $x=b$ directly, which isolates each constant in turn).

Worked reasoning

For $\frac{1}{(x+1)(x+2)}$, write $1=A(x+2)+B(x+1)$. Setting $x=-1$: $1=A(1) \Rightarrow A=1$. Setting $x=-2$: $1=B(-1) \Rightarrow B=-1$. So

and integrating term-by-term gives $\ln|x+1|-\ln|x+2|+C=\ln\left|\frac{x+1}{x+2}\right|+C$.