Integration Techniques

Replace a sub-expression with $u$.

$\int 2x(x^2+1)^3\,dx$: let $u = x^2+1$, $du = 2x\,dx$. $\int u^3\,du = \frac{u^4}{4}+C$.

$\int u\,dv = uv - \int v\,du$

$\int xe^x\,dx$: $u=x, dv=e^x\,dx$ → $xe^x - e^x + C$.

$\int \frac{1}{(x-1)(x+2)}\,dx = \int \frac{1/3}{x-1} - \frac{1/3}{x+2}\,dx = \frac{1}{3}\ln|x-1| - \frac{1}{3}\ln|x+2| + C$.