# 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)$