Antiderivative

Antidifferentiation (or "Indefinite integration") is a branch 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 as opposed to a value (of area under a graph, for example).

Antidifferentiation 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 simply means "with respect to $x$", which has no significance with simple antidifferentiation.

Finding a simple antiderivative

To find the antiderivative of a simple equation $ax^{n}$, the power $n$ should be increased by 1, then the whole equation should be divided by the new power, and then 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 consisting of several $x$ power terms can also be found in the same way, by finding the antiderivative of each section separately, 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 of simple antidifferentiation

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

Changing fractions and roots into powers makes the process easier.

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

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

Antiderivative of $e$

Antidifferentiating the exponential constant e is different; the result is still $e$:

$\int e\ dx=e$

However, if $e$ has a power of an equation such as $5x^{2}$, then the result is the original equation times the differential of the power of $e$:

$\int e^{5x^{2}}\ dx=e^{5x^{2}}\times 10x=10xe^{5x^{2}}$

Because differentiating $5x^{2}$ gives $10x$. As a general rule:

$\int e^{equation}=(e^{equation}\times \mathbf {differential\ of\ equation} )+c$