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I have the following (m and n are integers) $$ f[t] =\frac{ m^2n(n-1)}{2}t^2 \left(1 - (1 - t)^m \right)^{(n - 2)} (1 - t)^{(m - 1)} $$ $$ \text{Integrate}[f[t], {t, 0, 1}]$$ I also need the aymptotic case of the integral when m goes to infinity .Could somebody kindly help me this .Any help will be greatly appreciated .Presently I do not have access to Mathematica

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The result of your integral is

(m*n)/2 + Gamma[1 + n]*(-(Gamma[1/m]/Gamma[1/m + n]) + Gamma[2/m]/Gamma[2/m + n])

and the limit for m-> Infinity is 0

Edit:

use

Integrate[((m^2*n*(n - 1))/2)*t^2*(1 - (1 - t)^m)^(n - 2)*(1 - t)^(m - 1), t]

and insert the integration limits t->1 and t->0. Result is confirmed numerically.

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  • $\begingroup$ @ Andreas could you kindly provide the code as well so that I can run it on wolfram cloud free account $\endgroup$ May 2, 2021 at 8:37
  • $\begingroup$ fyi, limit t->0 seems to hang on V 12.2, or take longer time than I could wait. $\endgroup$
    – Nasser
    May 2, 2021 at 9:58
  • $\begingroup$ @nasser just insert t=0 $\endgroup$
    – Andreas
    May 2, 2021 at 10:05

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