The discussion focuses on the convergence of the integral $\int_{0}^{\infty} \frac{-\ln \left[ \frac{1}{a} x ^{ \frac{2}{a-1} } e ^{- x^{\frac{2}{a} }} \right] }{1+x^2} \mbox{d}x$ for values of $a \ge 2$. It is established that the integral converges for $a > 2$ and diverges when $a = 2$. The analysis involves rewriting the integral as $I = \int_{0}^{\infty} f(x)\ d x = \int_{0}^{\infty} \frac{\ln a - \frac{2}{a-1}\ \ln x + x^{\frac{2}{a}}}{1 + x^{2}}\ dx$, leading to the asymptotic behavior of $f(x)$ as $x$ approaches infinity.
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maxkor said:Investigate the convergence of the integral
$\int_{0}^{\infty} \frac{-\ln \left[ \frac{1}{a} x ^{ \frac{2}{a-1} } e ^{- x^{\frac{2}{a} }} \right] }{1+x^2} \mbox{d}x$ for $a \ge 2$