Integral of $$$3 e^{3 x}$$$

Find $$$\int 3 e^{3 x}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = e^{3 x}$$$:

$${\color{red}{\int{3 e^{3 x} d x}}} = {\color{red}{\left(3 \int{e^{3 x} d x}\right)}}$$

Let $$$u=3 x$$$.

Then $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{3}$$$.

Thus,

$$3 {\color{red}{\int{e^{3 x} d x}}} = 3 {\color{red}{\int{\frac{e^{u}}{3} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$$3 {\color{red}{\int{\frac{e^{u}}{3} d u}}} = 3 {\color{red}{\left(\frac{\int{e^{u} d u}}{3}\right)}}$$

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

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

Recall that $$$u=3 x$$$:

$$e^{{\color{red}{u}}} = e^{{\color{red}{\left(3 x\right)}}}$$

Therefore,

$$\int{3 e^{3 x} d x} = e^{3 x}$$

Add the constant of integration:

$$\int{3 e^{3 x} d x} = e^{3 x}+C$$

$$$\int 3 e^{3 x}\, dx = e^{3 x} + C$$$A