Integral rules cheat sheet

Integration is pattern matching against a short table plus two transformations. Substitution and parts, that reshape an integral until it matches the table.

Practise integrals

Standard antiderivatives

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
$\int e^{x} d x = e^{x} + C$
$\int a^{x} d x = \frac{a ^{x}}{l n a} + C$
$\int sin x d x = - cos x + C$
$\int sec^{2} x d x = tan x + C$
$\int \frac{d x}{1 + x ^{2}} = arctan x + C$

Techniques

$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ Substitution
$\int u d v = uv - \int v d u$ By parts
$\int \frac{P ( x )}{Q ( x )} d x$ Partial fractions when deg P < deg Q

Definite integrals

$\int_{a}^{b} f = F (b) - F (a)$ Fundamental theorem
$\int_{a}^{b} f = - \int_{b}^{a} f$
$\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$
$\frac{d}{d x} \int_{a}^{x} f (t) d t = f (x)$

Applications

$A = \int_{a}^{b} (f - g) d x$ Area between curves
$V = π \int_{a}^{b} f (x)^{2} d x$ Volume, discs
$f ˉ = \frac{1}{b - a} \int_{a}^{b} f$ Average value

Related calculators

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