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

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

Sea $$$u=\cos{\left(x \right)}$$$.

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

La integral puede reescribirse como

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = e^{u}$$$:

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

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

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

Recordemos que $$$u=\cos{\left(x \right)}$$$:

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

Por lo tanto,

$$\int{e^{\cos{\left(x \right)}} \sin{\left(x \right)} d x} = - e^{\cos{\left(x \right)}}$$

Añade la constante de integración:

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

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