Integralen av $$$\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}}$$$

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

Skriv om integranden:

$${\color{red}{\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x}}} = {\color{red}{\int{\frac{1}{2} d x}}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=\frac{1}{2}$$$:

$${\color{red}{\int{\frac{1}{2} d x}}} = {\color{red}{\left(\frac{x}{2}\right)}}$$

Alltså,

$$\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x} = \frac{x}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x} = \frac{x}{2}+C$$

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