AHL 5.16 (HL)—Advanced integration techniques

Advanced integration techniques expose hidden structure.

Substitution reverses a chain rule. Integration by parts reverses a product rule: $\int u\,dv=uv-\int v\,du$; repeat it when the remaining integral still contains a product suited to the same process.

Example

For $\int xe^x\,dx$, choose $u=x$ and $dv=e^x dx$. Then $du=dx$, $v=e^x$, so the integral is $xe^x-\int e^x dx=e^x(x-1)+C$.

Use substitution when one factor is the derivative of an inner function; use parts when differentiating one factor simplifies it. Transform limits as well as the differential in a definite substitution.

Partial fractions belong to AHL 5.15. In this objective, do not mix them into the choice between substitution and integration by parts.