特殊な形の積分

$$\int_{-\infty}^{\infty}e^{-ax^2}dx = \sqrt{\frac{\pi}{a}}$$ (5.13)

$$\int_{0}^{\infty}e^{-ax^2}dx = \frac{1}{2} \sqrt{\frac{\pi}{a}}$$ (5.14)

$$\int_{-\infty}^{\infty}e^{-ax^2}x^2dx = \frac{1}{2a}\sqrt{\frac{\pi}{a}}$$ (5.15)

$$\int_{0}^{\infty}e^{-ax^2}x^2dx = \frac{1}{4a}\sqrt{\frac{\pi}{a}}$$ (5.16)

$$\int_{-\infty}^{\infty}e^{-ax^2}x^{s}dx = \frac{1}{2}\left(1+(-1)^s\right) a^{-\frac{1}{2}(1+s)} \Gamma\left(\frac{1+s}{2}\right)$$ (5.17)

$$\int_{0}^{\infty}e^{-ax^2}x^{s}dx = \frac{1}{2} a^{-\frac{1}{2}(1+s)} \Gamma\left(\frac{1+s}{2}\right)$$ (5.18)

$$\int_{\alpha}^{\beta} (x-\alpha)^m(\beta-x)^n dx = \frac{m! n!}{(m+n+1)!}(\beta-\alpha)^{m+n+1}$$ (5.19)