Antiderivative

Antidifferentiation (or indefinite integration) is a part of mathematics, which is the opposite of differentiation. It is called indefinite integration because an equation is integrated without limits, so the result is an equation.

It is written as $\int x\ dx$

• with the integral sign that has no limits $\int$

• the equation you are integrating $x$

• and the $dx$ which means "with respect to $x$", which is not important with simple integration.

Finding a simple antiderivative

To find the antiderivative of a simple equation $ax^{n}$

• the power $n$ should be increased by 1
• the whole equation should be divided by the new power
• and a constant $c$ should be added (unlike definite integration).

This can be shown as:

$\int ax^{n}\ dx={\frac {ax^{n+1}}{n+1}}\ +\ c$

The antiderivative of an equation with several $x$ power terms can also be found in the same way, by finding the antiderivative of each section on its own, and then adding or subtracting.

$\int 2x^{6}-5x^{4}\ dx={\frac {2x^{7}}{7}}-{\frac {5x^{5}}{5}}+c={\frac {2}{7}}x^{7}-x^{5}+c$

Examples

$\int 3x^{4}\ dx={\frac {3x^{5}}{5}}\ +\ c$

Changing fractions and roots into powers makes it easier:

$\int {\frac {1}{x^{3}}}\ dx=\int x^{-3}\ dx={\frac {x^{-2}}{-2}}+c=-{\frac {1}{2x^{2}}}+c$

$\int {\sqrt {x^{3}}}\ dx=\int x^{\frac {3}{2}}\ dx={\frac {x^{\frac {5}{2}}}{\frac {5}{2}}}+c={\frac {2}{5}}x^{\frac {5}{2}}+c={\sqrt {{\frac {2}{5}}x^{5}}}+c$