Integrals
- $a$, $b$, $c$, $\alpha$, $\beta$ are constants
- $f$, $g$, $u$ are functions of a variable $x$
Basic properties
Properties
-
$$ \int _a ^a f(x)dx = 0 $$
-
$$ \int _b ^a f(x)dx = - \int _a ^b f(x)dx $$
-
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
$$ \int \frac {u'} {u} dx = ln |u| + c $$
$$ \int \frac {-u'} {u^2} dx = \frac {1} {u} + c $$
Common primitive functions
$f(x)$ | Domain $I$ | $F(x)$ |
---|---|---|
$x^n$ | $\Bbb{R}$ | $\frac{x^{n+1}}{n+1}$ |
$1/x$ | $]0,+\infty[$ | $\ln x$ |
$\ln x$ | $]0,+\infty[$ | $x\ln x -x$ |
$e^x$ | $\Bbb{R}$ | $e^x$ |
$\cos x$ | $\Bbb{R}$ | $\sin x$ |
$\sin x$ | $\Bbb{R}$ | $-\cos x$ |
$\tan x$ | $\Bbb{R}$ | $-\ln \mid\cos x\mid$ |
$\sinh x$ | $\Bbb{R}$ | $\cosh x$ |
$\cosh x$ | $\Bbb{R}$ | $\sinh x$ |
$\tanh x$ | $\Bbb{R}$ | $\ln(\cosh x)$ |