Integraal van $$$\frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}}$$$

Bepaal $$$\int \frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}}\, dx$$$.

Zij $$$u=8 x$$$.

Dan $$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{8}$$$.

Dus,

$${\color{red}{\int{\frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}} d x}}} = {\color{red}{\int{\frac{\sin{\left(2 u \right)}}{8 \cos{\left(u \right)}} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{8}$$$ en $$$f{\left(u \right)} = \frac{\sin{\left(2 u \right)}}{\cos{\left(u \right)}}$$$:

$${\color{red}{\int{\frac{\sin{\left(2 u \right)}}{8 \cos{\left(u \right)}} d u}}} = {\color{red}{\left(\frac{\int{\frac{\sin{\left(2 u \right)}}{\cos{\left(u \right)}} d u}}{8}\right)}}$$

Herschrijf de integraand:

$$\frac{{\color{red}{\int{\frac{\sin{\left(2 u \right)}}{\cos{\left(u \right)}} d u}}}}{8} = \frac{{\color{red}{\int{2 \sin{\left(u \right)} d u}}}}{8}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=2$$$ en $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{2 \sin{\left(u \right)} d u}}}}{8} = \frac{{\color{red}{\left(2 \int{\sin{\left(u \right)} d u}\right)}}}{8}$$

De integraal van de sinus is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}$$

We herinneren eraan dat $$$u=8 x$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{4} = - \frac{\cos{\left({\color{red}{\left(8 x\right)}} \right)}}{4}$$

Dus,

$$\int{\frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}} d x} = - \frac{\cos{\left(8 x \right)}}{4}$$

Voeg de integratieconstante toe:

$$\int{\frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}} d x} = - \frac{\cos{\left(8 x \right)}}{4}+C$$

$$$\int \frac{\sin{\left(16 x \right)}}{\cos{\left(8 x \right)}}\, dx = - \frac{\cos{\left(8 x \right)}}{4} + C$$$A