Integral: Investigating Convergence I

[maxkor]

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 $a \ge 2$

 

[chisigma]

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 $a \ge 2$

If You write the integral as ...

$\displaystyle I = \int_{0}^{\infty} f(x)\ d x = \int_{0}^{\infty} \frac{\ln a - \frac{2}{a-1}\ \ln x + x^{\frac{2}{a}}}{1 + x^{2}}\ dx\ (1)$

... in x tends to $\infty$ You have $\displaystyle f(x) \sim \frac{x^{\frac{2}{a}}}{1 + x^{2}}$, so that the integral converges for a>2 and diverges for a=2...

Kind regards

$\chi$ $\sigma$