Integration 2 Topic Review summary

Key learning points

• Integration by parts: $\int u\,dv=uv-\int v\,du$.
• Trig integrals: reduce powers with identities first.
• Rational functions: partial fractions then integrate.
• Always differentiate to verify multi-step antiderivatives.
• Definite integrals: substitute limits carefully after the antiderivative.

Worked example 1 — By parts

$\int x e^x\,dx$.

Working

$u=x$, $dv=e^x dx$.

Worked example 2 — PF integrate

$\int\frac{1}{x(x+1)}\,dx$.

Working

Decompose first.

Worked example 3 — Integration by parts: \\int u\\,dv=uv-\\in

Integration by parts: $\\int u\\,dv=uv-\\int v\\,du$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Trig integrals: reduce powers with identit

Apply this idea from the Topic Review: Trig integrals: reduce powers with identities first.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.