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Integrals

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$a$, $b$, $c$, $\alpha$, $\beta$ 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

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

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