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Integration
A-Level H2 Math
Trigonometric Integrals
$\int \tan x\, dx = \ln|\sec x| + C$
$\int \sec x\, dx = \ln|\sec x + \tan x| + C$
$\int \csc x\, dx = -\ln|\csc x + \cot x| + C$
$\int \cot x\, dx = \ln|\sin x| + C$
Half-angle identities
$\int \sin^2 x\, dx = \int \dfrac{1-\cos 2x}{2}\, dx$
$\int \cos^2 x\, dx = \int \dfrac{1+\cos 2x}{2}\, dx$
$\int \tan^2 x\, dx = \int (\sec^2 x - 1)\, dx$
$\int \cot^2 x\, dx = \int (\csc^2 x - 1)\, dx$
Direct results
$\int \sec^2 x\, dx = \tan x + C$
$\int \csc^2 x\, dx = -\cot x + C$
Given in MF27
Standard Form I
$\int [f(x)]^n f'(x)\, dx$$\int \dfrac{f'(x)}{f(x)}\, dx$$\int e^{f(x)}f'(x)\, dx$
$= \dfrac{[f(x)]^{n+1}}{n+1} + C$$= \ln|f(x)| + C$$= e^{f(x)} + C$
Standard Form II — MF27
$f(x)$$\dfrac{1}{x^2+a^2}$$\dfrac{1}{\sqrt{a^2-x^2}}$$\dfrac{1}{x^2-a^2}$$\dfrac{1}{a^2-x^2}$
$\int f(x)\,dx$$\dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right)$$\sin^{-1}\!\left(\dfrac{x}{a}\right)$$\dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right|$$\dfrac{1}{2a}\ln\left|\dfrac{a+x}{a-x}\right|$
Integration by Parts
$\int u\, dv = uv - \int v\, du$
LIATE Rule — choose $u$ in this priority:
LLogarithmic→
IInverse trig→
AAlgebraic→
TTrigonometric→
EExponential
Mnemonic: IK − ∫ID (I keep minus I differentiate)

Formulas for the Singapore A-Level H2 Mathematics syllabus