Integraal van $$$b d m o \sin{\left(x \right)} \cos{\left(x \right)}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int b d m o \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=b d m o$$$ en $$$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$$$:

$${\color{red}{\int{b d m o \sin{\left(x \right)} \cos{\left(x \right)} d x}}} = {\color{red}{b d m o \int{\sin{\left(x \right)} \cos{\left(x \right)} d x}}}$$

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

$$b d m o {\color{red}{\int{\sin{\left(x \right)} \cos{\left(x \right)} d x}}} = b d m o {\color{red}{\int{u d u}}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

$$b d m o {\color{red}{\int{u d u}}}=b d m o {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}=b d m o {\color{red}{\left(\frac{u^{2}}{2}\right)}}$$

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

$$\frac{b d m o {\color{red}{u}}^{2}}{2} = \frac{b d m o {\color{red}{\sin{\left(x \right)}}}^{2}}{2}$$

Dus,

$$\int{b d m o \sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{b d m o \sin^{2}{\left(x \right)}}{2}$$

Voeg de integratieconstante toe:

$$\int{b d m o \sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{b d m o \sin^{2}{\left(x \right)}}{2}+C$$

$$$\int b d m o \sin{\left(x \right)} \cos{\left(x \right)}\, dx = \frac{b d m o \sin^{2}{\left(x \right)}}{2} + C$$$A