Existence of the integral \(\displaystyle{\int_{{\alpha}}^{{\beta}}}\sqrt{{{\frac{{{\arctan{{h}}}{\left({r}\right)}}}{{{r}}}}}}{d}{r}\) For \(\displaystyle-{1}\le\alpha\le\beta\le{1}\)

Erik Cantu

Erik Cantu

Answered question


Existence of the integral αβarctanh(r)rdr
For 1αβ1

Answer & Explanation



Beginner2022-04-03Added 13 answers

arctanh rr is an even function, so it is enough to show that
0tarctanh  rrdr
is finite for any t[0,1]. This can be done through the Cauchy-Schwarz inequality, since
01arctanh  rrdr=π28
0tarctanh  rrdr0t1 dr0tarctanh  rrdrπt8
We may also state 0tarctanh  rrdrπt22 by exploiting convexity, but the above (quite crude) inequality is already enough to prove that αβarctanh  rrdr is finite for any α,β[1,1]

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