ガウス積分

\[\int_{0}^{\infty}x^{\gamma}e^{-\alpha x^{b}}dx=\frac{\alpha^{-\frac{1+\gamma}{b}}}{1+\gamma}\Gamma\left(\frac{1+\gamma}{b}+1\right)\]

\[\int_{0}^{\infty}e^{-\alpha x^{b}}dx=\frac{1}{\alpha^{\frac{1}{b}}}\Gamma\left(\frac{1}{b}+1\right)\]

\[\int_{0}^{\infty}e^{-\alpha x^{2}}dx=\frac{1}{2}\sqrt{\frac{\pi}{\alpha}}\]

\[\int_{0}^{x}\sin\left(x^{2}\right)dx=\sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{x^{4k+3}}{\left(2k+1\right)!\left(4k+3\right)}\]

\[\int_{0}^{\infty}\sin\left(x^{a}\right)dx=\Gamma\left(\frac{1}{a}+1\right)\sin\frac{\pi}{2a}\]

\[\int_{0}^{\infty}\sin\left(x^{2}\right)dx=\frac{\sqrt{2\pi}}{4}\]

\[S\left(x\right):=\int_{0}^{x}\sin\left(x^{2}\right)dx\]