Integral of $$$e^{\frac{x}{5}}$$$

Find $$$\int e^{\frac{x}{5}}\, dx$$$.

Let $$$u=\frac{x}{5}$$$.

Then $$$du=\left(\frac{x}{5}\right)^{\prime }dx = \frac{dx}{5}$$$ (steps can be seen »), and we have that $$$dx = 5 du$$$.

Thus,

$${\color{red}{\int{e^{\frac{x}{5}} d x}}} = {\color{red}{\int{5 e^{u} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=5$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{5 e^{u} d u}}} = {\color{red}{\left(5 \int{e^{u} d u}\right)}}$$

The integral of the exponential function is $$$\int{e^{u} d u} = e^{u}$$$:

$$5 {\color{red}{\int{e^{u} d u}}} = 5 {\color{red}{e^{u}}}$$

Recall that $$$u=\frac{x}{5}$$$:

$$5 e^{{\color{red}{u}}} = 5 e^{{\color{red}{\left(\frac{x}{5}\right)}}}$$

Therefore,

$$\int{e^{\frac{x}{5}} d x} = 5 e^{\frac{x}{5}}$$

Add the constant of integration:

$$\int{e^{\frac{x}{5}} d x} = 5 e^{\frac{x}{5}}+C$$

$$$\int e^{\frac{x}{5}}\, dx = 5 e^{\frac{x}{5}} + C$$$A