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

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

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

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

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

Entonces,

$${\color{red}{\int{e^{x} \ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot e^{x}-\int{e^{x} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(e^{x} \ln{\left(x \right)} - \int{\frac{e^{x}}{x} d x}\right)}}$$

Esta integral (Integral exponencial) no tiene una forma cerrada:

$$e^{x} \ln{\left(x \right)} - {\color{red}{\int{\frac{e^{x}}{x} d x}}} = e^{x} \ln{\left(x \right)} - {\color{red}{\operatorname{Ei}{\left(x \right)}}}$$

Por lo tanto,

$$\int{e^{x} \ln{\left(x \right)} d x} = e^{x} \ln{\left(x \right)} - \operatorname{Ei}{\left(x \right)}$$

Añade la constante de integración:

$$\int{e^{x} \ln{\left(x \right)} d x} = e^{x} \ln{\left(x \right)} - \operatorname{Ei}{\left(x \right)}+C$$

$$$\int e^{x} \ln\left(x\right)\, dx = \left(e^{x} \ln\left(x\right) - \operatorname{Ei}{\left(x \right)}\right) + C$$$A