I'm trying to do an integral, in which the integrand is composed of arbitrary functions and contains a derivative of the variable of integration. Further still, the integral is over a discontinuity. Because these are arbitrary functions I assume there is no exact solution, but I'm looking for an approximation.(adsbygoogle = window.adsbygoogle || []).push({});

Anyway, there is a discontinuity at 'x'

[tex]

\lim_{\Delta x \to \infty} \int_{x-\Delta x}^{x+\Delta x} \frac{1}{g(r)} f'(r) dr

[/tex]

The functions are defined in the limits on each side of the discontinuity, i.e.

[tex]

\lim_{\Delta x \to 0} \hspace{0.3in} f(x+\Delta x) = A \hspace{0.3in} f(x-\Delta x) = B \hspace{0.3in}

g(x+\Delta x) = C \hspace{0.3in} g(x-\Delta x) = D

[/tex]

Thus, if the integrand did not contain g(r), the result would be simple (from Leibniz's rule)

[tex]

\lim_{\Delta x \to \infty} \int_{x-\Delta x}^{x+\Delta x} f'(r) dr = A - B

[/tex]

Any help, tips, or pointers would be greatly appreciated!

Thanks,

Z

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# Approximation of Integral of arbitrary functions, containing a deriv, cross discont

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