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

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

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

Então $$$du=\left(\cos{\left(y \right)}\right)^{\prime }dy = - \sin{\left(y \right)} dy$$$ (veja os passos »), e obtemos $$$\sin{\left(y \right)} dy = - du$$$.

Portanto,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-1$$$ e $$$f{\left(u \right)} = e^{u}$$$:

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

A integral da função exponencial é $$$\int{e^{u} d u} = e^{u}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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