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

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

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

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

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

La integral se convierte en

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=-1$$$ y $$$f{\left(x \right)} = \sin{\left(\ln{\left(x \right)} \right)}$$$:

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

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

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

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

La integral puede reescribirse como

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

Hemos llegado a una integral que ya hemos visto.

Así, hemos obtenido la siguiente ecuación simple con respecto a la integral:

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

Al resolverlo, obtenemos que

$$\int{\cos{\left(\ln{\left(x \right)} \right)} d x} = \frac{x \left(\sin{\left(\ln{\left(x \right)} \right)} + \cos{\left(\ln{\left(x \right)} \right)}\right)}{2}$$

Por lo tanto,

$$\int{\cos{\left(\ln{\left(x \right)} \right)} d x} = \frac{x \left(\sin{\left(\ln{\left(x \right)} \right)} + \cos{\left(\ln{\left(x \right)} \right)}\right)}{2}$$

Simplificar:

$$\int{\cos{\left(\ln{\left(x \right)} \right)} d x} = \frac{\sqrt{2} x \sin{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}$$

Añade la constante de integración:

$$\int{\cos{\left(\ln{\left(x \right)} \right)} d x} = \frac{\sqrt{2} x \sin{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+C$$

$$$\int \cos{\left(\ln\left(x\right) \right)}\, dx = \frac{\sqrt{2} x \sin{\left(\ln\left(x\right) + \frac{\pi}{4} \right)}}{2} + C$$$A