Integrals

• $a$, $b$, $c$, $\alpha$, $\beta$ are constants
• $f$, $g$, $u$ are functions of a variable $x$

Basic properties

Properties

1. $\int _a ^a f(x)dx = 0$
2. $\int _b ^a f(x)dx = - \int _a ^b f(x)dx$
3. Chasles' theorem: $\int _a ^b f(x)dx + \int _b ^c f(x)dx = \int _a ^c f(x)dx$

Linearity

$\int (\alpha f(x) + \beta g(x)) dx = \alpha \int f(x) dx + \beta \int g(x) dx$

Integration by substitution

$\int _a ^b (f \circ g) (t) g'(t) dt = \int _{g(a)} ^{g(b)} f(x) dx$ when substituting $x = g(t)$

Integration by parts

$\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx$

Common integrals

Polynomials

$\int dx = x +c$

$\int x^{n}dx = \frac {1}{n+1} x^{n+1}+c$

Others

$\int \frac {u'} {u} dx = ln |u| + c$

$\int \frac {-u'} {u^2} dx = \frac {1} {u} + c$