Indefinite Integration

Standard integrals

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1),$ $\int \frac{d x}{x} = ln ∣ x ∣ + C,$ $\int e^{x} d x = e^{x} + C,$ $\int a^{x} d x = \frac{a ^{x}}{ln a} + C$
$\int tan x d x = ln ∣ sec x ∣ + C,$ $\int sec x d x = ln ∣ sec x + tan x ∣ + C,$ $\int csc x d x = ln ∣ csc x - cot x ∣ + C$

$\int \frac{d x}{x ^{2} + a ^{2}} = \frac{1}{a} tan^{- 1} \frac{x}{a} + C,$ $\int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} \frac{x}{a} + C$
$\int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{x - a}{x + a} + C,$ $\int \frac{d x}{a ^{2} - x ^{2}} = \frac{1}{2 a} ln \frac{a + x}{a - x} + C$
$\int \frac{d x}{x ^{2} \pm a ^{2}} = ln x + x^{2} \pm a^{2} + C$
$\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a} + C$
$\int x^{2} \pm a^{2} d x = \frac{x}{2} x^{2} \pm a^{2} \pm \frac{a ^{2}}{2} ln x + x^{2} \pm a^{2} + C$
$\int e^{a x} sin b x d x = \frac{e ^{a x}}{a ^{2} + b ^{2}} (a sin b x - b cos b x) + C$

By parts
$\int u v d x = u \int v d x - \int (u^{'} \int v d x) d x$
Choose $u$ in the order ILATE: inverse trig, log, algebraic, trig, exponential.
$\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) + C$
Substitution
$\int f (g (x)) g^{'} (x) d x = \int f (t) d t,$ $t = g (x)$
$\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣ + C,$ $\int \frac{f ^{'} ( x )}{f ( x )} d x = 2 f (x) + C$
Rational functions
$\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}$
Use partial fractions after making the numerator's degree less than the denominator's.
Trig form
$\int \frac{d x}{a + b cos x} : t = tan \frac{x}{2}, cos x = \frac{1 - t ^{2}}{1 + t ^{2}}, d x = \frac{2 d t}{1 + t ^{2}}$