Integralen av $$$\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$$ med avseende på $$$x$$$

Bestäm $$$\int \frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$$ och $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x}}} = {\color{red}{\left(\frac{\int{\sin{\left(5 x \right)} d x}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\right)}}$$

Låt $$$u=5 x$$$ vara.

Då $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{5}$$$.

Integralen blir

$$\frac{{\color{red}{\int{\sin{\left(5 x \right)} d x}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{5}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$

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

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$

Kom ihåg att $$$u=5 x$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} = - \frac{\cos{\left({\color{red}{\left(5 x\right)}} \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$

Alltså,

$$\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x} = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x} = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}+C$$

$$$\int \frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\, dx = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} + C$$$A