Integral of $$$960 e^{\frac{x}{120}}$$$

Find $$$\int 960 e^{\frac{x}{120}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=960$$$ and $$$f{\left(x \right)} = e^{\frac{x}{120}}$$$:

$${\color{red}{\int{960 e^{\frac{x}{120}} d x}}} = {\color{red}{\left(960 \int{e^{\frac{x}{120}} d x}\right)}}$$

Let $$$u=\frac{x}{120}$$$.

Then $$$du=\left(\frac{x}{120}\right)^{\prime }dx = \frac{dx}{120}$$$ (steps can be seen »), and we have that $$$dx = 120 du$$$.

The integral becomes

$$960 {\color{red}{\int{e^{\frac{x}{120}} d x}}} = 960 {\color{red}{\int{120 e^{u} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=120$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$$960 {\color{red}{\int{120 e^{u} d u}}} = 960 {\color{red}{\left(120 \int{e^{u} d u}\right)}}$$

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

$$115200 {\color{red}{\int{e^{u} d u}}} = 115200 {\color{red}{e^{u}}}$$

Recall that $$$u=\frac{x}{120}$$$:

$$115200 e^{{\color{red}{u}}} = 115200 e^{{\color{red}{\left(\frac{x}{120}\right)}}}$$

Therefore,

$$\int{960 e^{\frac{x}{120}} d x} = 115200 e^{\frac{x}{120}}$$

Add the constant of integration:

$$\int{960 e^{\frac{x}{120}} d x} = 115200 e^{\frac{x}{120}}+C$$

$$$\int 960 e^{\frac{x}{120}}\, dx = 115200 e^{\frac{x}{120}} + C$$$A