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

Find $$$\int u^{2} \cos{\left(2 u \right)}\, du$$$.

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

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

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

The integral becomes

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

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

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

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

Therefore,

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

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 $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$:

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

Let $$$v=2 u$$$.

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

The integral can be rewritten as

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

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

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

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

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

Recall that $$$v=2 u$$$:

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

Therefore,

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

Add the constant of integration:

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

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