$$$e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$$ 的積分

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

令 $$$u=\sin{\left(x \right)}$$$。

則 $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(x \right)} dx = du$$$。

所以，

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

指數函數的積分為 $$$\int{e^{u} d u} = e^{u}$$$：

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

回顧一下 $$$u=\sin{\left(x \right)}$$$：

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

因此，

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

加上積分常數：

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

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