Integraal van $$$\frac{\pi t \cos{\left(n \right)}}{2}$$$ met betrekking tot $$$t$$$

Bepaal $$$\int \frac{\pi t \cos{\left(n \right)}}{2}\, dt$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ toe met $$$c=\frac{\pi \cos{\left(n \right)}}{2}$$$ en $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{\frac{\pi t \cos{\left(n \right)}}{2} d t}}} = {\color{red}{\left(\frac{\pi \cos{\left(n \right)} \int{t d t}}{2}\right)}}$$

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

$$\frac{\pi \cos{\left(n \right)} {\color{red}{\int{t d t}}}}{2}=\frac{\pi \cos{\left(n \right)} {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}}{2}=\frac{\pi \cos{\left(n \right)} {\color{red}{\left(\frac{t^{2}}{2}\right)}}}{2}$$

Dus,

$$\int{\frac{\pi t \cos{\left(n \right)}}{2} d t} = \frac{\pi t^{2} \cos{\left(n \right)}}{4}$$

Voeg de integratieconstante toe:

$$\int{\frac{\pi t \cos{\left(n \right)}}{2} d t} = \frac{\pi t^{2} \cos{\left(n \right)}}{4}+C$$

$$$\int \frac{\pi t \cos{\left(n \right)}}{2}\, dt = \frac{\pi t^{2} \cos{\left(n \right)}}{4} + C$$$A