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Integrals

March 13, 2020

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

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

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

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)$