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

Bestäm $$$\int \frac{\sin{\left(2 z \right)}}{z}\, dz$$$.

Låt $$$u=2 z$$$ vara.

Då $$$du=\left(2 z\right)^{\prime }dz = 2 dz$$$ (stegen kan ses »), och vi har att $$$dz = \frac{du}{2}$$$.

Alltså,

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

Denna integral (Sinusintegralen) har ingen sluten form:

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

Kom ihåg att $$$u=2 z$$$:

$$\operatorname{Si}{\left({\color{red}{u}} \right)} = \operatorname{Si}{\left({\color{red}{\left(2 z\right)}} \right)}$$

Alltså,

$$\int{\frac{\sin{\left(2 z \right)}}{z} d z} = \operatorname{Si}{\left(2 z \right)}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sin{\left(2 z \right)}}{z} d z} = \operatorname{Si}{\left(2 z \right)}+C$$

$$$\int \frac{\sin{\left(2 z \right)}}{z}\, dz = \operatorname{Si}{\left(2 z \right)} + C$$$A