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Integralen av $$$\sin{\left(t^{2} \right)}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \sin{\left(t^{2} \right)}\, dt$$$.
Lösning
Denna integral (Fresnels sinusintegral) har ingen sluten form:
$${\color{red}{\int{\sin{\left(t^{2} \right)} d t}}} = {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} t}{\sqrt{\pi}}\right)}{2}\right)}}$$
Alltså,
$$\int{\sin{\left(t^{2} \right)} d t} = \frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} t}{\sqrt{\pi}}\right)}{2}$$
Lägg till integrationskonstanten:
$$\int{\sin{\left(t^{2} \right)} d t} = \frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} t}{\sqrt{\pi}}\right)}{2}+C$$
Svar
$$$\int \sin{\left(t^{2} \right)}\, dt = \frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} t}{\sqrt{\pi}}\right)}{2} + C$$$A