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

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

Para la integral $$$\int{x \cos{\left(x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=x$$$ y $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$.

Entonces $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$ (los pasos pueden verse »).

Por lo tanto,

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

La integral del seno es $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

$$$\int x \cos{\left(x \right)}\, dx = \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) + C$$$A