Integration of Exponential Functions

Formula 1:

$$\int e^x \, dx = e^x+C $$

Proof:

\begin{align*} \because \frac{d}{dx} \, e^x &= e^x \\\\ \frac{d}{dx} \, (e^x+C) &= e^x \\\\ \therefore \int e^x \,dx &= e^x +C \\\\ \end{align*}

Example 1:

$$\int e^{-x} \, dx $$

Solution:

$$\int e^{-x} \, dx $$ \begin{align*} \textup{Let}\,\, u&=-x\\ du&=-dx\\ -du&=dx\\ \int e^{-x} \, dx &= \int e^u\,\, (-du)\\ &= -\int e^u\,du\\ &= -\, e^u+C\\ &= -\, e^{-x}+C\\ \end{align*}

Example 2:

$$\int e^{4x} \, dx $$

Solution:

$$\int e^{4x} \, dx $$ \begin{align*} \textup{Let}\,\, u & =4x \\ du & =4dx \\ \frac{du}4 & =dx \\\\ \int e^{4x} \, dx &= \int e^u\,\, \frac{du}4 \\ & = \frac{1}4 \int e^u\,du \\ & = \frac{1}4 \, e^u+C \\ & = \frac{1}4 \, e^{4x}+C \\ \end{align*}

Example 3:

$$\int e^{5x+2} \, dx $$

Solution:

$$\int e^{5x+2} \, dx $$ \begin{align*} \textup{Let}\,\, u & =5x+2 \\ du & =5dx \\ \frac{du}5 & =dx \\\\ \int e^{5x+2} \, dx &= \int e^u\,\, \frac{du}5 \\ & = \frac{1}5 \int e^u\,du \\ & = \frac{1}5 \, e^u+C \\ & = \frac{1}5 \, e^{5x+2}+C \\ \end{align*}