Integraal van $$$81 \cos^{2}{\left(x \right)}$$$

Bepaal $$$\int 81 \cos^{2}{\left(x \right)}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=81$$$ en $$$f{\left(x \right)} = \cos^{2}{\left(x \right)}$$$:

$${\color{red}{\int{81 \cos^{2}{\left(x \right)} d x}}} = {\color{red}{\left(81 \int{\cos^{2}{\left(x \right)} d x}\right)}}$$

Pas de machtsreductieformule $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ toe met $$$\alpha=x$$$:

$$81 {\color{red}{\int{\cos^{2}{\left(x \right)} d x}}} = 81 {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}}$$

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 $$$f{\left(x \right)} = \cos{\left(2 x \right)} + 1$$$:

$$81 {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}} = 81 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}{2}\right)}}$$

Integreer termgewijs:

$$\frac{81 {\color{red}{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}}}{2} = \frac{81 {\color{red}{\left(\int{1 d x} + \int{\cos{\left(2 x \right)} d x}\right)}}}{2}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$\frac{81 \int{\cos{\left(2 x \right)} d x}}{2} + \frac{81 {\color{red}{\int{1 d x}}}}{2} = \frac{81 \int{\cos{\left(2 x \right)} d x}}{2} + \frac{81 {\color{red}{x}}}{2}$$

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

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

De integraal kan worden herschreven als

$$\frac{81 x}{2} + \frac{81 {\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{2} = \frac{81 x}{2} + \frac{81 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$

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

$$\frac{81 x}{2} + \frac{81 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = \frac{81 x}{2} + \frac{81 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$

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

$$\frac{81 x}{2} + \frac{81 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{81 x}{2} + \frac{81 {\color{red}{\sin{\left(u \right)}}}}{4}$$

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

$$\frac{81 x}{2} + \frac{81 \sin{\left({\color{red}{u}} \right)}}{4} = \frac{81 x}{2} + \frac{81 \sin{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

Dus,

$$\int{81 \cos^{2}{\left(x \right)} d x} = \frac{81 x}{2} + \frac{81 \sin{\left(2 x \right)}}{4}$$

Vereenvoudig:

$$\int{81 \cos^{2}{\left(x \right)} d x} = \frac{81 \left(2 x + \sin{\left(2 x \right)}\right)}{4}$$

Voeg de integratieconstante toe:

$$\int{81 \cos^{2}{\left(x \right)} d x} = \frac{81 \left(2 x + \sin{\left(2 x \right)}\right)}{4}+C$$

$$$\int 81 \cos^{2}{\left(x \right)}\, dx = \frac{81 \left(2 x + \sin{\left(2 x \right)}\right)}{4} + C$$$A