Integral of $$$96 x^{2} \cos{\left(2 x \right)}$$$

Find $$$\int 96 x^{2} \cos{\left(2 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=96$$$ and $$$f{\left(x \right)} = x^{2} \cos{\left(2 x \right)}$$$:

$${\color{red}{\int{96 x^{2} \cos{\left(2 x \right)} d x}}} = {\color{red}{\left(96 \int{x^{2} \cos{\left(2 x \right)} d x}\right)}}$$

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

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

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

So,

$$96 {\color{red}{\int{x^{2} \cos{\left(2 x \right)} d x}}}=96 {\color{red}{\left(x^{2} \cdot \frac{\sin{\left(2 x \right)}}{2}-\int{\frac{\sin{\left(2 x \right)}}{2} \cdot 2 x d x}\right)}}=96 {\color{red}{\left(\frac{x^{2} \sin{\left(2 x \right)}}{2} - \int{x \sin{\left(2 x \right)} d x}\right)}}$$

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

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

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (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 »).

Thus,

$$48 x^{2} \sin{\left(2 x \right)} - 96 {\color{red}{\int{x \sin{\left(2 x \right)} d x}}}=48 x^{2} \sin{\left(2 x \right)} - 96 {\color{red}{\left(x \cdot \left(- \frac{\cos{\left(2 x \right)}}{2}\right)-\int{\left(- \frac{\cos{\left(2 x \right)}}{2}\right) \cdot 1 d x}\right)}}=48 x^{2} \sin{\left(2 x \right)} - 96 {\color{red}{\left(- \frac{x \cos{\left(2 x \right)}}{2} - \int{\left(- \frac{\cos{\left(2 x \right)}}{2}\right)d x}\right)}}$$

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

$$48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} + 96 {\color{red}{\int{\left(- \frac{\cos{\left(2 x \right)}}{2}\right)d x}}} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} + 96 {\color{red}{\left(- \frac{\int{\cos{\left(2 x \right)} d x}}{2}\right)}}$$

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

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

So,

$$48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 48 {\color{red}{\int{\cos{\left(2 x \right)} d x}}} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 48 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

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

$$48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 48 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 48 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 {\color{red}{\sin{\left(u \right)}}}$$

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

$$48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 \sin{\left({\color{red}{u}} \right)} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 \sin{\left({\color{red}{\left(2 x\right)}} \right)}$$

Therefore,

$$\int{96 x^{2} \cos{\left(2 x \right)} d x} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 \sin{\left(2 x \right)}$$

Add the constant of integration:

$$\int{96 x^{2} \cos{\left(2 x \right)} d x} = 48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 \sin{\left(2 x \right)}+C$$

$$$\int 96 x^{2} \cos{\left(2 x \right)}\, dx = \left(48 x^{2} \sin{\left(2 x \right)} + 48 x \cos{\left(2 x \right)} - 24 \sin{\left(2 x \right)}\right) + C$$$A