Integraal van $$$\cos{\left(3 x \right)} \cos{\left(5 x \right)}$$$

Bepaal $$$\int \cos{\left(3 x \right)} \cos{\left(5 x \right)}\, dx$$$.

Herschrijf de integraand met behulp van de formule $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$, met $$$\alpha=3 x$$$ en $$$\beta=5 x$$$:

$${\color{red}{\int{\cos{\left(3 x \right)} \cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(8 x \right)}}{2}\right)d x}}}$$

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

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

Integreer termgewijs:

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

Zij $$$u=2 x$$$.

Dan $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{2}$$$.

Dus,

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

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

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

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

$$\frac{\int{\cos{\left(8 x \right)} d x}}{2} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{\int{\cos{\left(8 x \right)} d x}}{2} + \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$

We herinneren eraan dat $$$u=2 x$$$:

$$\frac{\int{\cos{\left(8 x \right)} d x}}{2} + \frac{\sin{\left({\color{red}{u}} \right)}}{4} = \frac{\int{\cos{\left(8 x \right)} d x}}{2} + \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

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,

$$\frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\int{\cos{\left(8 x \right)} d x}}}}{2} = \frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{2}$$

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{2} = \frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}}{2}$$

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

$$\frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{16} = \frac{\sin{\left(2 x \right)}}{4} + \frac{{\color{red}{\sin{\left(u \right)}}}}{16}$$

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

$$\frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left({\color{red}{u}} \right)}}{16} = \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left({\color{red}{\left(8 x\right)}} \right)}}{16}$$

Dus,

$$\int{\cos{\left(3 x \right)} \cos{\left(5 x \right)} d x} = \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(8 x \right)}}{16}$$

Voeg de integratieconstante toe:

$$\int{\cos{\left(3 x \right)} \cos{\left(5 x \right)} d x} = \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(8 x \right)}}{16}+C$$

$$$\int \cos{\left(3 x \right)} \cos{\left(5 x \right)}\, dx = \left(\frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(8 x \right)}}{16}\right) + C$$$A