Integration of Exponential Functions

The base formula of integrating exponential function is obtained from deriving $e^x$. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} $$ This base formula is extended to the following general formula. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} \times a \\ &= ae^{ax+b} \\ e^{ax+b} &= \int{ae^{ax+b}}dx \\ e^{ax+b} &= a\int{e^{ax+b}}dx \\ \dfrac{1}{a}e^{ax+b} &= \int{e^{ax+b}}dx \\ \therefore \int{e^{ax+b}}dx &= \dfrac{1}{a}e^{ax+b} +c \end{align} $$

Example 1

Find $\displaystyle \int{2e^x}dx$.

Example 2

Find $\displaystyle \int{e^{2x+1}}dx$.

Example 3

Find $\displaystyle \int{\dfrac{1}{e^x}}dx$.

Example 4

Find $\displaystyle \int{\sqrt{e^x}}dx$.