Divergent Integrals

HYPERSECTIONS

Monomial Integrals

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

Polynomial Integrals

$\displaystyle \int_x^{\infty_+} (a+bt) \, dt = -a+b\left(\dfrac{1}{2}-x\right) = -a + \dfrac{b}{2} - b\,x$

$\displaystyle \int_x^{\infty_+} (a+bt+ct^2) \, dt = -a + b\left(\dfrac{1}{2}-x\right) - c\left(\dfrac{1}{3} - x + x^2\right) \\ = -a+\dfrac{b}{2}-\dfrac{c}{3} - (b-c)\,x - c\,x^2$

$\displaystyle \int_x^{\infty_+} (a+bt+ct^2+dt^3) \, dt \\ = -a + b\left(\dfrac{1}{2}-x\right) - c\left(\dfrac{1}{3}-x+x^2\right) + d\left(\dfrac{1}{4}-x+\dfrac{3}{2}x^2-x^3\right) \\ = -a+\dfrac{b}{2}-\dfrac{c}{3}+\dfrac{d}{4} - (b-c+d)\,x - \left(c-\dfrac{3d}{2}\right)\,x^2 - d\,x^3$

Exponential Integrals

$\displaystyle \int_x^{\infty_+} e^{at} \, dt = -\dfrac{e^{ax}}{a}$

$\displaystyle \int_x^{\infty_+} e^{at+b} \, dt = -\dfrac{e^{ax+b}}{a}$

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

$\displaystyle \int_x^{\infty_+} a^{bt} \, dt = -\dfrac{a^{bx}}{b\ln a}$

Trigonometric Integrals

$\displaystyle \int_x^{\infty_+} e^{iat} \, dt = i \, \dfrac{e^{iax}}{a} = -\dfrac{\sin(ax)}{a} \,+\, i\,\dfrac{\cos(ax)}{a}$

$\displaystyle \int_x^{\infty_+} \cos(at) \, dt = -\dfrac{\sin(ax)}{a}$

$\displaystyle \int_x^{\infty_+} \sin(at) \, dt = \dfrac{\cos(ax)}{a}$

$\displaystyle \int_x^{\infty_+} e^{i(at\!+\!b)} \, dt = i \, \dfrac{e^{i(ax+b)}}{a} = -\dfrac{\sin(ax\!+\!b)}{a} \,+\, i\,\dfrac{\cos(ax\!+\!b)}{a}$

$\displaystyle \int_x^{\infty_+} \cos(at\!+\!b) \, dt = -\dfrac{\sin(ax\!+\!b)}{a}$

$\displaystyle \int_x^{\infty_+} \sin(at\!+\!b) \, dt = \dfrac{\cos(ax\!+\!b)}{a}$

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