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

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

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).

So,

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

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

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

Therefore,

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

Add the constant of integration:

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

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