Integral of $$$3 x^{4} e^{x^{5}}$$$

Find $$$\int 3 x^{4} e^{x^{5}}\, dx$$$.

Let $$$u=x^{5}$$$.

Then $$$du=\left(x^{5}\right)^{\prime }dx = 5 x^{4} dx$$$ (steps can be seen »), and we have that $$$x^{4} dx = \frac{du}{5}$$$.

The integral can be rewritten as

$${\color{red}{\int{3 x^{4} e^{x^{5}} d x}}} = {\color{red}{\int{\frac{3 e^{u}}{5} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{3}{5}$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{3 e^{u}}{5} d u}}} = {\color{red}{\left(\frac{3 \int{e^{u} d u}}{5}\right)}}$$

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

$$\frac{3 {\color{red}{\int{e^{u} d u}}}}{5} = \frac{3 {\color{red}{e^{u}}}}{5}$$

Recall that $$$u=x^{5}$$$:

$$\frac{3 e^{{\color{red}{u}}}}{5} = \frac{3 e^{{\color{red}{x^{5}}}}}{5}$$

Therefore,

$$\int{3 x^{4} e^{x^{5}} d x} = \frac{3 e^{x^{5}}}{5}$$

Add the constant of integration:

$$\int{3 x^{4} e^{x^{5}} d x} = \frac{3 e^{x^{5}}}{5}+C$$

$$$\int 3 x^{4} e^{x^{5}}\, dx = \frac{3 e^{x^{5}}}{5} + C$$$A