Revision sheet

Integration by parts

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

Watch out: the minus sign, and signs inside $\int v\,du$.

Choosing $u$

• LIATE: logs, inverse trig, powers, trig, exponentials.
• $x^n e^x$: parts $n$ times.

Watch out: if the new integral is harder, swap $u$ and $dv$.

Partial fractions

1. Factor the denominator.
2. Find $A$, $B$ by cover-up.
3. Integrate: $\displaystyle\int \dfrac{A}{x - r}\,dx = A\ln|x - r|$.

Watch out: check by recombining the fractions.

Which technique?

• Inside + its derivative: substitution.
• Unrelated product: parts.
• Factorable denominator: partial fractions.

Watch out: $x^2 + a^2$ (arctan) and $x^2 - a^2$ (fractions) differ.