Integral of $$$x^{5} \sin{\left(4 x^{6} \right)}$$$

Find $$$\int x^{5} \sin{\left(4 x^{6} \right)}\, dx$$$.

Let $$$u=4 x^{6}$$$.

Then $$$du=\left(4 x^{6}\right)^{\prime }dx = 24 x^{5} dx$$$ (steps can be seen »), and we have that $$$x^{5} dx = \frac{du}{24}$$$.

The integral becomes

$${\color{red}{\int{x^{5} \sin{\left(4 x^{6} \right)} d x}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{24} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{24}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{24} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{24}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{24} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{24}$$

Recall that $$$u=4 x^{6}$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{24} = - \frac{\cos{\left({\color{red}{\left(4 x^{6}\right)}} \right)}}{24}$$

Therefore,

$$\int{x^{5} \sin{\left(4 x^{6} \right)} d x} = - \frac{\cos{\left(4 x^{6} \right)}}{24}$$

Add the constant of integration:

$$\int{x^{5} \sin{\left(4 x^{6} \right)} d x} = - \frac{\cos{\left(4 x^{6} \right)}}{24}+C$$

$$$\int x^{5} \sin{\left(4 x^{6} \right)}\, dx = - \frac{\cos{\left(4 x^{6} \right)}}{24} + C$$$A