Mingwu Bai writes
> Please kindly advise me how to solve indefinite integral:
>
> int [(r^3 dr)/(sqrt(r^2+a^2)-b)]
Not sure whether you're asking for a numerical method or a symbolic
solution. The former is more appropriate for a Fortran mail list.
If you don't have access to a standard numerical library,
Simpson's rule would be adequate if not going too near a zero
of the denominator. (Obviously, the denominator may or may not
hav real zeros, depending on the relative sizes of a and b.)
The symbolic question is not really appropriate for this list
but is quite easy to answer. The substitution
u^2 = r^2 + a^2, giving u du = r dr
transforms the integral to
int [((u^2 - a^2) u du) / (u-b)].
The integrand can be divided out to give
polynomial + constant/(u-b),
and this when integrated with respect to u gives
polynomial + constant * log(u-b).
I've tried it with Mathematica 3 and Maple V rel 4
and they both (especially Maple) make it seem complicated, bringing
arctanh into it; but as there are expressions for inverse hyperbolics
in terms of logs, their answers are probably equivalent to the above.
Robert Hill
Information Systems Services
University of Leeds, UK
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