Integral of $$$x^{2} \cos{\left(x \right)}$$$

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

For the integral $$$\int{x^{2} \cos{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x^{2}$$$ and $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$.

Then $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$ (steps can be seen »).

So,

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

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

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

For the integral $$$\int{x \sin{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x$$$ and $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$.

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (steps can be seen »).

The integral becomes

$$x^{2} \sin{\left(x \right)} - 2 {\color{red}{\int{x \sin{\left(x \right)} d x}}}=x^{2} \sin{\left(x \right)} - 2 {\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}=x^{2} \sin{\left(x \right)} - 2 {\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

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

The integral of the cosine is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Therefore,

$$\int{x^{2} \cos{\left(x \right)} d x} = x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}$$

Add the constant of integration:

$$\int{x^{2} \cos{\left(x \right)} d x} = x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}+C$$

$$$\int x^{2} \cos{\left(x \right)}\, dx = \left(x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}\right) + C$$$A