I try to compute the value of the following integral $\int_0^{\frac{1}{b}}r^2(\frac{1}{r}-b)^{\frac{5}{2}}dr$ for b>0 Can you help me?
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2026-05-06 08:54:45.1778057685
Integration over a $(0,1/b)$
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Your integral looks very similar to the Beta Function$$B(m,n)=\int\limits_0^1dt\, t^{m-1}(1-t)^{n-1}$$In order to get there, we first need to make the expression in the parenthesis equal to $1-t$. The first step is to factor out a $\frac 1r$ to get$$I=\int\limits_0^{1/b}dr\, r^{-1/2}(1-br)^{5/2}$$Now make a substitution $t=br$. Can you complete the rest?