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

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

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

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

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

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = x \cos{\left(2 x \right)}$$$:

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

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

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

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\cos{\left(2 x \right)} d x}=\frac{\sin{\left(2 x \right)}}{2}$$$ (steps can be seen »).

So,

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$:

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

Let $$$u=2 x$$$.

Then $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{2}$$$.

Therefore,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

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

$$- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4} = - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}$$

Recall that $$$u=2 x$$$:

$$- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{u}} \right)}}{4} = - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

Therefore,

$$\int{x^{2} \sin{\left(2 x \right)} d x} = - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}$$

Add the constant of integration:

$$\int{x^{2} \sin{\left(2 x \right)} d x} = - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}+C$$

$$$\int x^{2} \sin{\left(2 x \right)}\, dx = \left(- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right) + C$$$A