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

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

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

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

Entonces,

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

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 $$$u=\cos{\left(y \right)}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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