Antiderivatives

Key results

The general antiderivative

$$G'(x) = F'(x) \text{ for all } x \text{ in an interval} \implies G(x) = F(x) + C$$$G^{\prime }(x)$ $=F^{\prime }(x)\text{ for all }x\text{ in an interval}$ $⟹G(x)$ $=F(x)+C$

For some constant $C$, on an interval only; on a domain made of separate intervals, such as $x \neq 0$, each interval gets its own constant.

Powers

$$f(x) = x^{n}:\quad F(x) = \frac{x^{n+1}}{n + 1} + C$$$f(x)$ $=x^n:$$F(x)$ $=\frac{x^{n+1}}{n+1}+C$

For any constant $n \neq -1$, on an interval where $x^{n}$ is defined.

The reciprocal

$$f(x) = \frac{1}{x}:\quad F(x) = \log|x| + C$$$f(x)$ $=\frac{1}{x}:$$F(x)$ $=\log |x|+C$

On $x > 0$ or on $x < 0$, one interval at a time.

Exponentials and trigonometric functions

$$\begin{array}{c|c} f(x) & F(x) \\ \hline e^{ax} & \frac{1}{a}e^{ax} + C \\ \cos(ax) & \frac{1}{a}\sin(ax) + C \\ \sin(ax) & -\frac{1}{a}\cos(ax) + C \\ \frac{1}{1 + x^{2}} & \arctan x + C \end{array}$$

For a constant $a \neq 0$, with $x$ in radians.

Constant acceleration

$$v(t) = v_{0} + a t, \qquad s(t) = s_{0} + v_{0} t + \tfrac{1}{2} a t^{2}$$$v(t)$ $=v_0+at,$$s(t)$ $=s_0+v_{0}t+\frac{1}{2}a{t}^2$

For a constant acceleration $a$, with $v_{0} = v(0)$ and $s_{0} = s(0)$.

·Read the theory·CLP-1 §4.1 Introduction to antiderivatives