Divergent Integrals

HYPERSPECIES

Monomial Integrals

$\displaystyle \int_x^{\infty_+} dt = -1$

$\displaystyle \int_x^{\infty_+} \sqrt{t} \, dt = \dfrac{2}{3} \left((x-1)^\frac{3}{2} - x^\frac{3}{2} \right)$

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

$\displaystyle \int_x^{\infty_+} t^2 \, dt = -\dfrac{1}{3} + x - x^2$

$\displaystyle \int_x^{\infty_+} t^3 \, dt = \dfrac{1}{4} - x + \dfrac{3x^2}{2} - x^3$

$\displaystyle \int_x^{\infty_+} t^4 \, dt = -\dfrac{1}{5} + x - 2x^2 + 2x^3 - x^4$

$\displaystyle \int_x^{\infty_+} t^5 \, dt = \dfrac{1}{6} - x + \dfrac{5x^2}{2} - \dfrac{10x^3}{3} + \dfrac{5x^4}{2} - x^5$

$\displaystyle \int_x^{\infty_+} t^6 \, dt = -\dfrac{1}{7} + x - 3x^2 + 5x^3 - 5x^4 + 3x^5 - x^6$

$\displaystyle \int_x^{\infty_+} t^7 \, dt = \dfrac{1}{8} - x + \dfrac{7x^2}{2} - 7x^3 + \dfrac{35x^4}{4} - 7x^5 + \dfrac{7x^6}{2} - x^7$

$\displaystyle \int_x^{\infty_+} t^8 \, dt = -\dfrac{1}{9} + x - 4x^2 + \dfrac{28x^3}{3} - 14x^4 + 14x^5 - \dfrac{28x^6}{3} + 4x^7 - x^8$

$\displaystyle \int_x^{\infty_+} t^9 \, dt = \dfrac{1}{10} - x + \dfrac{9x^2}{2} - 12x^3 + 21x^4 - \dfrac{126x^5}{5} + 21x^6 - 12x^7 + \dfrac{9x^8}{2} - x^9$

Monomial Ratio Integrals

$\displaystyle \int_x^{\infty_+} \dfrac{1}{t} \, dt = -\ln x$

Polynomial Integrals

$\displaystyle \int_x^{\infty_+} (1+t) \, dt = -\dfrac{1}{2} - x$

$\displaystyle \int_x^{\infty_+} (1+t+t^2) \, dt = -\dfrac{5}{6} - x^2$

$\displaystyle \int_x^{\infty_+} (1+t+t^2+t^3) \, dt = -\dfrac{7}{12} - x + \dfrac{x^2}{2} - x^3$

Polynomial Ratio Integrals

$\displaystyle \int_x^{\infty_+} \dfrac{1}{1-t} \, dt = \ln(1-x) - \ln(2-x) = \ln\!\left(\dfrac{1-x}{2-x}\right) = \ln\!\left(\dfrac{x-1}{x-2}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{1}{1+t} \, dt = \ln x - \ln(1+x) = \ln\!\left(\dfrac{x}{1+x}\right) = \ln\!\left(1-\dfrac{1}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t}{1+t} \, dt = -1 - \ln\!\left(\dfrac{x}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t^2}{1+t} \, dt = \dfrac{3}{2} - x + \ln\!\left(\dfrac{x}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t^3}{1+t} \, dt = -\dfrac{11}{6} + 2x - x^2 - \ln\!\left(\dfrac{x}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t^4}{1+t} \, dt = \dfrac{25}{12} - 3x + \dfrac{5}{2} x^2 - x^3 + \ln\!\left(\dfrac{x}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t^5}{1+t} \, dt = -\dfrac{137}{60} + 4x - \dfrac{9}{2} x^2 + 3x^3 - x^4 - \ln\!\left(\dfrac{x}{1+x}\right)$

$\displaystyle \int_x^{\infty_+} \dfrac{t^6}{1+t} \, dt = \dfrac{49}{20} - 5x + 7x^2 - \dfrac{19}{3} x^3 + \dfrac{7}{2} x^4 - x^5 + \ln\!\left(\dfrac{x}{1+x}\right)$

Trigonometric Integrals

$\displaystyle \int_x^{\infty_+} \sin t \, dt = \cos x - \cos(x-1)$

$\displaystyle \int_x^{\infty_+} \cos t \, dt = \sin(x-1) - \sin x$

$\displaystyle \int_x^{\infty_+} \tan t \, dt = \ln(\cos x) - \ln(\cos(x-1)) = \ln\!\left(\dfrac{\cos x}{\cos(x-1)}\right)$

$\displaystyle \int_x^{\infty_+} \csc t \, dt = \ln\!\left(\tan\dfrac{x-1}{2}\right) - \ln\!\left(\tan\dfrac{x}{2}\right) = \ln\!\left(\dfrac{\tan\dfrac{x-1}{2}}{\tan\dfrac{x}{2}}\right)$

$\displaystyle \int_x^{\infty_+} \sec t \, dt = \ln\!\left(\tan\!\left(\dfrac{x-1}{2}+\dfrac{\pi}{4}\right)\right) - \ln\!\left(\tan\!\left(\dfrac{x}{2}+\dfrac{\pi}{4}\right)\right) \\ \\ = \ln\!\left(\dfrac{\tan\!\left(\dfrac{x-1}{2}+\dfrac{\pi}{4}\right)}{\tan\!\left(\dfrac{x}{2}+\dfrac{\pi}{4}\right)}\right)$

$\displaystyle \int_x^{\infty_+} \cot t \, dt = \ln(\sin(x-1)) - \ln(\sin x) = \ln\!\left(\dfrac{\sin(x-1)}{\sin x}\right)$

$\displaystyle \int_x^{\infty_+} \sin(2t) \, dt = \dfrac{1}{2} \left(\cos(2x) - \cos(2(x-1))\right)$

$\displaystyle \int_x^{\infty_+} \cos(2t) \, dt = \dfrac{1}{2} \left(\sin(2(x-1)) - \sin(2x)\right)$

$\displaystyle \int_x^{\infty_+} \tan(2t) \, dt = \dfrac{1}{2} \left(\ln(\cos(2x)) - \ln(\cos(2(x-1)))\right) = \dfrac{1}{2} \ln\!\left(\dfrac{\cos(2x)}{\cos(2(x-1))}\right)$

$\displaystyle \int_x^{\infty_+} \csc(2t) \, dt = \dfrac{1}{2} \left(\ln\!\left(\tan\!\left(x-\dfrac{1}{2}\right)\right) - \ln\!\left(\tan x\right)\right) = \dfrac{1}{2} \ln\!\left(\dfrac{\tan\!\left(x-\dfrac{1}{2}\right)}{\tan x}\right)$

$\displaystyle \int_x^{\infty_+} \sec(2t) \, dt = \dfrac{1}{2} \left(\ln\!\left(\tan\!\left(x-\dfrac{1}{2}+\dfrac{\pi}{4}\right)\right) - \ln\!\left(\tan\!\left(x+\dfrac{\pi}{4}\right)\right)\right) \\ \\ = \dfrac{1}{2} \ln\!\left(\dfrac{\tan\left(x-\dfrac{1}{2}+\dfrac{\pi}{4}\right)}{\tan\left(x+\dfrac{\pi}{4}\right)}\right)$

$\displaystyle \int_x^{\infty_+} \cot(2t) \, dt = \dfrac{1}{2} \left(\ln(\sin(2(x-1))) - \ln(\sin(2x))\right) = \dfrac{1}{2} \ln\!\left(\dfrac{\sin(2(x-1))}{\sin(2x)}\right)$

$\displaystyle \int_x^{\infty_+} t\sin t \, dt = \sin(x-1) - \sin x - (x-1)\cos(x-1) + x\cos x$

$\displaystyle \int_x^{\infty_+} t\cos t \, dt = (x-1)\sin(x-1) - x\sin x + \cos(x-1) - \cos x$

$\displaystyle \int_x^{\infty_+} \sin^2 t \, dt = -\dfrac{1}{2} - \dfrac{1}{4} \left(\sin(2(x-1)) - \sin(2x)\right)$

$\displaystyle \int_x^{\infty_+} \cos^2 t \, dt = -\dfrac{1}{2} + \dfrac{1}{4} \left(\sin(2(x-1)) - \sin(2x)\right)$

$\displaystyle \int_x^{\infty_+} \tan^2 t \, dt = \int_x^{\infty_+} \dfrac{\sin^2 t}{\cos^2 t} \, dt = 1 + \tan(x-1) - \tan x$

$\displaystyle \int_x^{\infty_+} \csc^2 t \, dt = \int_x^{\infty_+} \dfrac{1}{\sin^2 t} \, dt = \cot x - \cot(x-1)$

$\displaystyle \int_x^{\infty_+} \sec^2 t \, dt = \int_x^{\infty_+} \dfrac{1}{\cos^2 t} \, dt = \tan(x-1) - \tan x$

$\displaystyle \int_x^{\infty_+} \cot^2 t \, dt = \int_x^{\infty_+} \dfrac{\cos^2 t}{\sin^2 t} \, dt = 1 + \cot x - \cot(x-1)$

$\displaystyle \int_x^{\infty_+} \sin t \, \sec^2 t \, dt = \int_x^{\infty_+} \dfrac{\sin t}{\cos^2 t} \, dt = \sec(x-1) - \sec x$

$\displaystyle \int_x^{\infty_+} \cos t \, \csc^2 t \, dt = \int_x^{\infty_+} \dfrac{\cos t}{\sin^2 t} \, dt = \csc x - \csc(x-1)$

Exponential Integrals

$\displaystyle \int_x^{\infty_+} e^t \, dt = - e^x$

$\displaystyle \int_x^{\infty_+} t\,e^t \, dt = (x\!-\!2)\,e^{x\!-\!1} - (x\!-\!1)\,e^x$

Logarithmic Integrals

$\displaystyle \int_x^{\infty_+} \ln t \, dt = x(1-\ln x)$

$\displaystyle \int_x^{\infty_+} t\ln t \, dt = \dfrac{x^2}{4} (3 - 2\ln x) - x$

$\displaystyle \int_x^{\infty_+} t(\ln t\!-\!1) \, dt = \dfrac{x^2}{4} (3 - 2\ln x) - \dfrac{1}{2}$

Special Function Integrals

$\displaystyle \int_x^{\infty_+} \ln\Gamma(t) \, dt = \psi^{(-2)}(x\!-\!1) - \psi^{(-2)}(x)$

$\displaystyle \int_x^{\infty_+} \psi^{(0)}(t) \, dt = \psi^{(-1)}(x\!-\!1) - \psi^{(-1)}(x) = -\ln(x\!-\!1)$

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