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In this answer Integral representation of Euler's constant the guy transformed the integral (4) into (5). Is there a general theorem or rule to follow? I mean, how can I do it for any integral?

Mr. N
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In that case, they had an integral defined from $t=0$ to $\infty$, and then they performed a substitution where $t=ln(1/k)$. In that case, to get the new integral limits, you substitute for the lower and upper limits like so:

New lower limit: $t=ln(1/k) = 0$ and solve for $k$. The value is $1$

New upper limit: $t=ln(1/k) = \infty$ and solve for $k$. The value is $0$

I think that's all there is to it.

Que
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  • I did this, but when we substitute we also have to calculate $dt$. Thanks – Mr. N Jan 19 '20 at 21:51
  • So, in general, these are the steps. change the variable and solve for the new one. – Mr. N Jan 19 '20 at 21:51
  • Well, yes, of course you should change $dt$ to $dk$. You also have to solve the integral. I thought you were asking exclusively about how to change the limits once you change the variable. – Que Jan 19 '20 at 21:56