Integralen av $$$\sin{\left(\frac{\pi x}{30} \right)}$$$

Bestäm $$$\int \sin{\left(\frac{\pi x}{30} \right)}\, dx$$$.

Låt $$$u=\frac{\pi x}{30}$$$ vara.

Då $$$du=\left(\frac{\pi x}{30}\right)^{\prime }dx = \frac{\pi}{30} dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{30 du}{\pi}$$$.

Alltså,

$${\color{red}{\int{\sin{\left(\frac{\pi x}{30} \right)} d x}}} = {\color{red}{\int{\frac{30 \sin{\left(u \right)}}{\pi} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{30}{\pi}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{30 \sin{\left(u \right)}}{\pi} d u}}} = {\color{red}{\left(\frac{30 \int{\sin{\left(u \right)} d u}}{\pi}\right)}}$$

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

$$\frac{30 {\color{red}{\int{\sin{\left(u \right)} d u}}}}{\pi} = \frac{30 {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{\pi}$$

Kom ihåg att $$$u=\frac{\pi x}{30}$$$:

$$- \frac{30 \cos{\left({\color{red}{u}} \right)}}{\pi} = - \frac{30 \cos{\left({\color{red}{\left(\frac{\pi x}{30}\right)}} \right)}}{\pi}$$

Alltså,

$$\int{\sin{\left(\frac{\pi x}{30} \right)} d x} = - \frac{30 \cos{\left(\frac{\pi x}{30} \right)}}{\pi}$$

Lägg till integrationskonstanten:

$$\int{\sin{\left(\frac{\pi x}{30} \right)} d x} = - \frac{30 \cos{\left(\frac{\pi x}{30} \right)}}{\pi}+C$$

$$$\int \sin{\left(\frac{\pi x}{30} \right)}\, dx = - \frac{30 \cos{\left(\frac{\pi x}{30} \right)}}{\pi} + C$$$A