Math Formulas
Exponential Integrals Formulas
$ \int^\infty_0 e^{-ax} \cos bx \, dx = \dfrac{a}{a^2 + b^2} \\ $
$ \int^\infty_0 e^{-ax} \sin bx \, ~ dx = \dfrac{b}{a^2 + b^2} \\ $
$ \int^\infty_0 \dfrac{e^{-ax} \sin bx}{x} \,~ dx = \arctan \dfrac{b}{a} \\ $
$ \int^\infty_0 \dfrac{e^{-ax}-e^{-bx}}{x} ~dx = \ln \dfrac{b}{a} \\ $
$ \int^\infty_0 e^{-ax^2} \, dx = \dfrac{1}{2} \sqrt{ \dfrac{\pi}{a} } \\ $
$ \int^\infty_0 e^{-ax^2} \cos bx \, dx = \dfrac{1}{2} \sqrt{ \dfrac{\pi}{a} } e^{-\dfrac{b^2}{4a}} \\ $
$ \int^\infty_{-\infty} e^{-(ax^2+bx+c)} dx = \sqrt{\dfrac{\pi}{a}} e^\dfrac{b^2-4ac}{4a} \\ $
$ \int^\infty_0 x^n\,e^{-ax}~dx = \dfrac{\Gamma(n+1)}{a^{n+1}} \\ $
$ \int^\infty_0 x^m\,e^{-ax^2}dx = \dfrac{\Gamma\left(\dfrac{m+1}{2}\right)}{2a^{(m+1)/2}} \\ $
$ \int^\infty_0 e^{-\left(ax^2+b/x^2\right)} dx = \dfrac{1}{2}\sqrt{\dfrac{\pi}{a}}e^{-2\sqrt{ab}} \\ $
$ \int^\infty_0 \dfrac{x^{n-1}}{e^x-1}dx = \Gamma (n) \left( \dfrac{1}{1^n} + \dfrac{1}{2^n} + \dfrac{1}{3^n} + \cdots \right) \\ $
$ \int^\infty_0 \dfrac{x\,dx}{e^x+1} = \dfrac{\pi^2}{12} \\ $
$ \int^\infty_0 \dfrac{x^{n-1}}{e^x+1}~dx = \Gamma (n) \left( \dfrac{1}{1^n} - \dfrac{1}{2^n} + \dfrac{1}{3^n} - \cdots \right) \\ $
$ \int^\infty_0 \dfrac{\sin mx}{e^{2\pi x} - 1} ~dx = \dfrac{1}{4}~ \coth \dfrac{m}{2} - \dfrac{1}{2m} \\ $
$ \int^\infty_0 \left( \dfrac{1}{1+x} - e^{-x} \right) \dfrac{dx}{x} = \gamma \\ $
$ \int^\infty_0 \dfrac{e^{-x^2} - e^{-x}}{x} ~dx = \dfrac{1}{2} \gamma \\ $
$ \int^\infty_0 \left( \dfrac{1}{e^x-1} - \dfrac{e^{-x}}{x} \right) ~ dx = \gamma \\ $
$ \int^\infty_0 \dfrac{e^{-ax} - e^{-bx}}{x \sec (px)} ~ dx = \dfrac{1}{2} ~ \ln\left( \dfrac{b^2+p^2}{a^2+p^2}\right) \\ $
$ \int^\infty_0 \dfrac{e^{-ax} - e^{-bx}}{x \csc (px)} ~ dx = \arctan \dfrac{b}{p} - \arctan \dfrac{a}{p} \\ $
$ \int^\infty_0 \dfrac{e^{-ax}(1-\cos x)}{x^2} ~ dx = \mathrm{arccot}\,a - \dfrac{a}{2} ~ \ln (a^2 + 1) \\ $