Antiderivative

Antidifferentiation (or indefinite integration) is a part of mathematics. It is the opposite of differentiation. It is integrating with no limits. The answer 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 does not mean anything with simple integration.

Finding a simple antiderivative

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

• add 1 to the power $n$, so $ax^{n}$ is now $ax^{n+1}$
• divide all this by the new power, so it is now ${\frac {ax^{n+1}}{n+1}}$
• and a constant $c$ should be added.

This can be shown as:

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

When there is many $x$ terms, integrate each part on its own:

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

(This only works if the parts are being added or taken away.)

Examples

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

Changing fractions and roots into powers makes it easier:

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

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