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
\[\int {\left( {{e^x} + x} \right)dx} = {e^x} + \frac{1}{2}{x^2} + c\]