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

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

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

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

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

So,

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

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

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

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

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

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

The integral can be rewritten as

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

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

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

Therefore,

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

Add the constant of integration:

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

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