Integral of e Power X

Integration of e^x is another important formula of integral calculus; this integral belongs exponential formulas category 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 , so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, 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