# Integration

### Integral

###### Integral

$$\int x^n \: dx = \dfrac {1}{n + 1} \:.\: x^{n + 1} + c$$

$$\int (ax + b)^n \: dx = \dfrac 1a \:.\: \dfrac {1}{n + 1} \:.\: x^{n + 1} + c$$

$$\int k \: dx = kx + c$$, $$k$$ is a constant

Example 01

\begin{equation*} \begin{split} \int x^4 \: dx & = \frac {1}{4 + 1} \:.\: x^{4 + 1} + c \\\\ \int x^4 \: dx & = \frac {1}{5} x^{5} + c \end{split} \end{equation*}

Example 02

\begin{equation*} \begin{split} \int 3x^8 \: dx & = 3 \:.\: \frac {1}{8 + 1} \:.\: x^{8 + 1} + c \\\\ \int 3x^8 \: dx & = \frac {1}{3} x^{9} + c \end{split} \end{equation*}

Example 03

\begin{equation*} \begin{split} \int \sqrt{x} \: dx & = \int x^{\frac 12} \: dx \\\\ \int \sqrt{x} \: dx & = \frac {1}{\frac 12 + 1} \:.\: x^{\frac 12 + 1} + c \\\\ \int \sqrt{x} \: dx & = \frac {1}{\frac 32} \:.\: x^{\frac 12 + 1} + c \\\\ \int \sqrt{x} \: dx & = \frac {2}{3} x^{1 \frac 12} + c \\\\ \int \sqrt{x} \: dx & = \frac {2}{3} x \sqrt{x} + c \end{split} \end{equation*}

Example 04

\begin{equation*} \begin{split} \int \frac {1}{x^3} \: dx & = \int x^{-3} \: dx \\\\ \int \frac {1}{x^3} \: dx & = \frac {1}{-3 + 1} \:.\: x^{-3 + 1} + c \\\\ \int \frac {1}{x^3} \: dx & = -\frac {1}{2} x^{-2} + c \\\\ \int \frac {1}{x^3} \: dx & = -\frac {1}{2 x^2} + c \end{split} \end{equation*}

Example 05

\begin{equation*} \begin{split} & \int (2x - 10)^3\:dx \\\\ & \frac 12 \:.\: \frac {1}{3 + 1} \:.\: (2x - 10)^{3 + 1} + c \\\\ & \bbox[5px, border: 2px solid magenta] {\frac{1}{8} (2x-10)^4 +c} \end{split} \end{equation*}

Example 06

\begin{equation*} \begin{split} & \int \frac {dx}{(4x - 1)^3} \\\\ & \int (4x - 1)^{-3} \: dx \\\\ & \frac 14 \:.\: \frac {1}{-3 + 1} \:.\: (4x - 1)^{-3 + 1} + c \\\\ & - \frac{1}{8} (4x - 1)^{-2} +c \\\\ & \bbox[5px, border: 2px solid magenta] {\frac{- 1}{8 \:.\: (4x - 1)^{2}} +c} \end{split} \end{equation*}

Example 07

\begin{equation*} \begin{split} \int 4 \: dx & = 4x + c \end{split} \end{equation*}