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






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