# Antidifferentiation

If $F'(x)=f(x)$, then $\int f(x) \cdot dx = F(x) + c$

$$\int x^n \cdot dx = {x^{n+1} \over {n+1}} + c, \quad n \in \mathbb{N} \cup \{0\}$$

Rules:

$\int [f(x) \pm g(x)] \cdot dx = \int f(x) \cdot dx \pm \int g(x) \cdot dx$  
$\int kf(x) \cdot dx = k \int f(x) \cdot dx$, where $k \in \mathbb{R}$

## Applications of differentiation to kinematics

Kinematics - straight line motion of a particle

Instantaneous velocity - dx/dt

## Newton's method

$$x_{n+1}=x_n - {f(x_n) \over f^\prime(x_n)}$$

or

$$x_1=x_0 - {f(x_0) \over f^\prime(x_0)}$$