Antiderivatives

Primitive function

A function $F (x)$ is antiderivative or primitive functions of a function $f (x)$ on an interval $ℐ$ if $F (x)$ is differentiable on $ℐ$ and

F^{'} (x) = f (x)

for all values in $ℐ$ .

If $F_{1} (x)$ and $F_{2} (x)$ are both primitive functions of $f (x)$ on an interval $ℐ$ then

F_{1} (x) - F_{2} (x) = 𝒸

for some fixed constant $𝒸 \in ℝ$ .

The collection of all primitive functions of $f (x)$ on an interval is called the indefinite integral and denoted

\int f (x) d x

\int (α f (x) + β g (x)) d x = α \int f (x) + β \int g (x)

\begin{matrix} 1 & \int x^{n} d x & = & \frac{x^{n + 1}}{n + 1} + 𝒸 \\ 2 & \int \frac{1}{x} d x & = & ln | x | + 𝒸 \\ 3 & \int ℯ^{x} d x & = & ℯ^{x} + 𝒸 \\ 4 & \int sin x d x & = & - cos x + 𝒸 \\ 5 & \int cos x d x & = & sin x + 𝒸 \end{matrix}

\begin{matrix} 1 & \int \frac{1}{\sqrt{1 - x^{2}}} d x & = & arcsin x + 𝒸 \\ 2 & \int \frac{1}{1 + x^{2}} d x & = & arctan x + 𝒸 \end{matrix}

\begin{matrix} 1 & \int tan x d x & = & ln | \frac{1}{cos x} | + 𝒸 = ln | sec x | + 𝒸 \\ 2 & \int cot x d x & = & ln | sin x | + 𝒸 \\ 3 & \int sec x d x & = & ln | sec x + tan x | + 𝒸 \\ 4 & \int csc x d x & = & - ln | csc x + cot x | + 𝒸 \end{matrix}

\begin{matrix} \int \sqrt{x} (x - 2) d x \\ \int \frac{(x - 1)}{\sqrt{x}} d x \end{matrix}