# Integral of e Power X

The integration of $e^x$ is another important formula of integral calculus. This integral belongs to the exponential formulae and is one of the simplest formula of integration.

The integration of e power x is of the form
$\int {{e^x}dx = } {e^x} + c$

Now consider
$\frac{d}{{dx}}\left[ {{e^x} + c} \right] = \frac{d}{{dx}}{e^x} + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}{e^x} = {e^x}$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {{e^x} + c} \right] = {e^x} + 0 \\ \Rightarrow {e^x} = \frac{d}{{dx}}\left[ {{e^x} + c} \right] \\ \Rightarrow {e^x}dx = d\left[ {{e^x} + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{e^x}dx} = \int {d\left[ {{e^x} + c} \right]}$

Since integration and differentiation are reverse processes to each other, the integral sign $\int {}$ and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.
$\int {{e^x}dx} = {e^x} + c$

Example: Evaluate the integral $\int {\left( {{e^x} + x} \right)dx}$ with respect to $x$

We have integral $I = \int {\left( {{e^x} + x} \right)dx}$
$\int {\left( {{e^x} + x} \right)dx} = \int {{e^x}dx + \int {xdx} }$

Using integration of e power x, we have
$\int {\left( {{e^x} + x} \right)dx} = {e^x} + \frac{1}{2}{x^2} + c$