Integraal van $$$e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$$

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

Zij $$$u=\sin{\left(x \right)}$$$.

Dan $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$\cos{\left(x \right)} dx = du$$$.

De integraal kan worden herschreven als

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

De integraal van de exponentiële functie is $$$\int{e^{u} d u} = e^{u}$$$:

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

We herinneren eraan dat $$$u=\sin{\left(x \right)}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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