Integral of $$$e^{\frac{t}{50}}$$$

Find $$$\int e^{\frac{t}{50}}\, dt$$$.

Let $$$u=\frac{t}{50}$$$.

Then $$$du=\left(\frac{t}{50}\right)^{\prime }dt = \frac{dt}{50}$$$ (steps can be seen »), and we have that $$$dt = 50 du$$$.

Thus,

$${\color{red}{\int{e^{\frac{t}{50}} d t}}} = {\color{red}{\int{50 e^{u} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=50$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{50 e^{u} d u}}} = {\color{red}{\left(50 \int{e^{u} d u}\right)}}$$

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

$$50 {\color{red}{\int{e^{u} d u}}} = 50 {\color{red}{e^{u}}}$$

Recall that $$$u=\frac{t}{50}$$$:

$$50 e^{{\color{red}{u}}} = 50 e^{{\color{red}{\left(\frac{t}{50}\right)}}}$$

Therefore,

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}$$

Add the constant of integration:

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}+C$$

$$$\int e^{\frac{t}{50}}\, dt = 50 e^{\frac{t}{50}} + C$$$A