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