Integralen av $$$\sin{\left(3 x^{2} \right)}$$$

Bestäm $$$\int \sin{\left(3 x^{2} \right)}\, dx$$$.

Låt $$$u=\sqrt{3} x$$$ vara.

Då $$$du=\left(\sqrt{3} x\right)^{\prime }dx = \sqrt{3} dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{\sqrt{3} du}{3}$$$.

Integralen blir

$${\color{red}{\int{\sin{\left(3 x^{2} \right)} d x}}} = {\color{red}{\int{\frac{\sqrt{3} \sin{\left(u^{2} \right)}}{3} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{\sqrt{3}}{3}$$$ och $$$f{\left(u \right)} = \sin{\left(u^{2} \right)}$$$:

$${\color{red}{\int{\frac{\sqrt{3} \sin{\left(u^{2} \right)}}{3} d u}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\sin{\left(u^{2} \right)} d u}}{3}\right)}}$$

Denna integral (Fresnels sinusintegral) har ingen sluten form:

$$\frac{\sqrt{3} {\color{red}{\int{\sin{\left(u^{2} \right)} d u}}}}{3} = \frac{\sqrt{3} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} u}{\sqrt{\pi}}\right)}{2}\right)}}}{3}$$

Kom ihåg att $$$u=\sqrt{3} x$$$:

$$\frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{2} {\color{red}{u}}}{\sqrt{\pi}}\right)}{6} = \frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{2} {\color{red}{\sqrt{3} x}}}{\sqrt{\pi}}\right)}{6}$$

Alltså,

$$\int{\sin{\left(3 x^{2} \right)} d x} = \frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6}$$

Lägg till integrationskonstanten:

$$\int{\sin{\left(3 x^{2} \right)} d x} = \frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6}+C$$

$$$\int \sin{\left(3 x^{2} \right)}\, dx = \frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6} + C$$$A