Integralen av $$$\cos{\left(\pi t \right)}$$$

Bestäm $$$\int \cos{\left(\pi t \right)}\, dt$$$.

Låt $$$u=\pi t$$$ vara.

Då $$$du=\left(\pi t\right)^{\prime }dt = \pi dt$$$ (stegen kan ses »), och vi har att $$$dt = \frac{du}{\pi}$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{\pi}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

Integralen av cosinus är $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

Kom ihåg att $$$u=\pi t$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{\pi} = \frac{\sin{\left({\color{red}{\pi t}} \right)}}{\pi}$$

Alltså,

$$\int{\cos{\left(\pi t \right)} d t} = \frac{\sin{\left(\pi t \right)}}{\pi}$$

Lägg till integrationskonstanten:

$$\int{\cos{\left(\pi t \right)} d t} = \frac{\sin{\left(\pi t \right)}}{\pi}+C$$

$$$\int \cos{\left(\pi t \right)}\, dt = \frac{\sin{\left(\pi t \right)}}{\pi} + C$$$A