Integral of $$$x^{2} \cos{\left(t \right)}$$$ with respect to $$$x$$$

Find $$$\int x^{2} \cos{\left(t \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\cos{\left(t \right)}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{x^{2} \cos{\left(t \right)} d x}}} = {\color{red}{\cos{\left(t \right)} \int{x^{2} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$\cos{\left(t \right)} {\color{red}{\int{x^{2} d x}}}=\cos{\left(t \right)} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\cos{\left(t \right)} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{x^{2} \cos{\left(t \right)} d x} = \frac{x^{3} \cos{\left(t \right)}}{3}$$

Add the constant of integration:

$$\int{x^{2} \cos{\left(t \right)} d x} = \frac{x^{3} \cos{\left(t \right)}}{3}+C$$

$$$\int x^{2} \cos{\left(t \right)}\, dx = \frac{x^{3} \cos{\left(t \right)}}{3} + C$$$A