Revision sheet

Substitution

1. Choose the inside $u = g(x)$.
2. Replace $g'(x)\,dx$ by $du$ (adjust a constant).
3. Integrate in $u$, then return to $x$.

Watch out: keep track of constant factors from $du$.

Definite integrals by substitution

• New limits: $u(a)$ and $u(b)$.
• Integrate in $u$ between them.

Watch out: never mix $x$-limits with a $u$-integral.

Useful patterns

• $\dfrac{g'}{g} \to \ln|g|$.
• $e^{kx} \to \dfrac{e^{kx}}{k}$.
• $\tan x \to -\ln|\cos x|$.

Watch out: a minus sign from $du$.

Algebra first

• Divide term by term when the bottom is one term.
• Complete the square for $\dfrac{1}{x^2 + bx + c}$, then arctan.

Watch out: $\dfrac{1}{a + b} \ne \dfrac{1}{a} + \dfrac{1}{b}$.