Integral de $$$x \cos{\left(x \right)}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu aportación
Encuentra $$$\int x \cos{\left(x \right)}\, dx$$$.
Solución
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).
Thus,
$${\color{red}{\int{x \cos{\left(x \right)} d x}}}={\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}={\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$
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: $$$\int{x \cos{\left(x \right)} d x}=x \sin{\left(x \right)} + \cos{\left(x \right)}+C$$$