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

Bestäm $$$\int \frac{\sin{\left(t^{2} \right)}}{t}\, dt$$$.

Låt $$$u=t^{2}$$$ vara.

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

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \frac{\sin{\left(u \right)}}{u}$$$:

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

Denna integral (Sinusintegralen) har ingen sluten form:

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

Kom ihåg att $$$u=t^{2}$$$:

$$\frac{\operatorname{Si}{\left({\color{red}{u}} \right)}}{2} = \frac{\operatorname{Si}{\left({\color{red}{t^{2}}} \right)}}{2}$$

Alltså,

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

Lägg till integrationskonstanten:

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

$$$\int \frac{\sin{\left(t^{2} \right)}}{t}\, dt = \frac{\operatorname{Si}{\left(t^{2} \right)}}{2} + C$$$A