Integralen av $$$\sin{\left(\pi x \right)}$$$

Bestäm $$$\int \sin{\left(\pi x \right)}\, dx$$$.

Låt $$$u=\pi x$$$ vara.

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

Integralen blir

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

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

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

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

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

Kom ihåg att $$$u=\pi x$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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