Divergent Integrals

GENERA

Monomial Integrals

$\displaystyle \int_0^{\infty_+} t^n \, dt = \dfrac{(-1)^{n+1}}{n+1}$

$\displaystyle \int_0^{\infty_+} t^{2n-1} \, dt = \dfrac{1}{2n}$

$\displaystyle \int_0^{\infty_+} t^{2n} \, dt = -\dfrac{1}{2n+1}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^n}{n} \, dt = \dfrac{(-1)^{n+1}}{n(n+1)}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^n}{n!} \, dt = \dfrac{(-1)^{n+1}}{(n+1)!}$

$\displaystyle \int_0^{\infty_+} t^{r-1} \, dt = \dfrac{(-1)^r}{r}$

Polynomial Ratio Integrals

$\displaystyle \int_0^{\infty_+} \dfrac{t^n}{1+t} \, dt = (-1)^n \, (H_n - \gamma)$

$\displaystyle \int_0^1 \dfrac{t^n}{1-t} \, dt = \gamma - H_n$

$\displaystyle \int_0^1 \dfrac{t^n}{(1-t)^2} \, dt = \dfrac{1}{n+1} \; _2F_1(2,n\!+\!1;n\!+\!2;1) = -n\gamma - \dfrac{1}{2} + H_n^{(2)}$

$\displaystyle \int_0^1 \dfrac{t^m}{1+t^n} \, dt = \dfrac{1}{n} \; \Gamma\!\left(\dfrac{m+1}{n}\right) \; \Gamma\!\left(1-\dfrac{m+1}{n}\right) = \dfrac{\pi}{n} \; \csc\left(\dfrac{m+1}{n}\,\pi\right)$

$\displaystyle \int_0^{\infty_+} \dfrac{1}{t^{2n}\,(1+t^2)} \, dt = (-1)^n \, \left(\dfrac{\pi}{2} + \sum_{m=1}^n (-1)^m \, \dfrac{\gamma^{2m-1}}{2m-1}\right)$

$\displaystyle \int_0^{\infty_+} \dfrac{1}{t^{2n+1}\,(1+t^2)} \, dt = (-1)^n \, \left(\gamma + \sum_{m=1}^n (-1)^m \, \dfrac{\gamma^{2m}}{2m}\right)$

Laplace Transforms (s > 0)

Mellin Transforms (s > 0)

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^t \, dt = -\dfrac{1}{s} \int_0^{\infty_+} t^s \, e^t \, dt = \dfrac{\Gamma(s)}{(-1)^s}$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, \ln t \, dt = \dfrac{(-1)^{s-1}}{s^2} \,+\, i\pi \dfrac{(-1)^{s}}{s} = \dfrac{(-1)^{s-1}}{s^2}\,(1-i\pi s)$