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

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

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

Then $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (steps can be seen »), and we have that $$$\sin{\left(x \right)} dx = - du$$$.

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = e^{u}$$$:

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

The integral of the exponential function is $$$\int{e^{u} d u} = e^{u}$$$:

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

Recall that $$$u=\cos{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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