Integral de $$$x e^{\frac{x}{5}}$$$

Halla $$$\int x e^{\frac{x}{5}}\, dx$$$.

Para la integral $$$\int{x e^{\frac{x}{5}} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=x$$$ y $$$\operatorname{dv}=e^{\frac{x}{5}} dx$$$.

Entonces $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{e^{\frac{x}{5}} d x}=5 e^{\frac{x}{5}}$$$ (los pasos pueden verse »).

Por lo tanto,

$${\color{red}{\int{x e^{\frac{x}{5}} d x}}}={\color{red}{\left(x \cdot 5 e^{\frac{x}{5}}-\int{5 e^{\frac{x}{5}} \cdot 1 d x}\right)}}={\color{red}{\left(5 x e^{\frac{x}{5}} - \int{5 e^{\frac{x}{5}} d x}\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=5$$$ y $$$f{\left(x \right)} = e^{\frac{x}{5}}$$$:

$$5 x e^{\frac{x}{5}} - {\color{red}{\int{5 e^{\frac{x}{5}} d x}}} = 5 x e^{\frac{x}{5}} - {\color{red}{\left(5 \int{e^{\frac{x}{5}} d x}\right)}}$$

Sea $$$u=\frac{x}{5}$$$.

Entonces $$$du=\left(\frac{x}{5}\right)^{\prime }dx = \frac{dx}{5}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = 5 du$$$.

La integral puede reescribirse como

$$5 x e^{\frac{x}{5}} - 5 {\color{red}{\int{e^{\frac{x}{5}} d x}}} = 5 x e^{\frac{x}{5}} - 5 {\color{red}{\int{5 e^{u} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=5$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$$5 x e^{\frac{x}{5}} - 5 {\color{red}{\int{5 e^{u} d u}}} = 5 x e^{\frac{x}{5}} - 5 {\color{red}{\left(5 \int{e^{u} d u}\right)}}$$

La integral de la función exponencial es $$$\int{e^{u} d u} = e^{u}$$$:

$$5 x e^{\frac{x}{5}} - 25 {\color{red}{\int{e^{u} d u}}} = 5 x e^{\frac{x}{5}} - 25 {\color{red}{e^{u}}}$$

Recordemos que $$$u=\frac{x}{5}$$$:

$$5 x e^{\frac{x}{5}} - 25 e^{{\color{red}{u}}} = 5 x e^{\frac{x}{5}} - 25 e^{{\color{red}{\left(\frac{x}{5}\right)}}}$$

Por lo tanto,

$$\int{x e^{\frac{x}{5}} d x} = 5 x e^{\frac{x}{5}} - 25 e^{\frac{x}{5}}$$

Simplificar:

$$\int{x e^{\frac{x}{5}} d x} = 5 \left(x - 5\right) e^{\frac{x}{5}}$$

Añade la constante de integración:

$$\int{x e^{\frac{x}{5}} d x} = 5 \left(x - 5\right) e^{\frac{x}{5}}+C$$

$$$\int x e^{\frac{x}{5}}\, dx = 5 \left(x - 5\right) e^{\frac{x}{5}} + C$$$A