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Hernaner28
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Homework Statement
(a). Prove that the improper integrals converge:
[tex] \displaystyle \int\limits_{0}^{1}{\frac{\ln x}{1+{{x}^{2}}}}dx[/tex]
[tex] \displaystyle \int\limits_{1}^{\infty }{\frac{\ln x}{1+{{x}^{2}}}}dx[/tex]
And relate each other.
(b) Deduce the value of:
[tex] \displaystyle \int\limits_{0}^{\infty }{\frac{\ln x}{1+{{x}^{2}}}}dx[/tex]
Homework Equations
The Attempt at a Solution
I managed to prove that [tex] \displaystyle \int\limits_{0}^{1}{\frac{\ln x}{1+{{x}^{2}}}}dx[/tex] converges by bounding it by [tex] \displaystyle \frac{1}{\sqrt{x}}[/tex] which converges. Now, I cannot bound the other integral so I applied integration by parts and obtained:
[tex] \displaystyle \ln x\cdot \arctan x\left. \begin{align}
& \\
& \\
\end{align} \right|_{1}^{\infty }-\int\limits_{1}^{\infty }{\frac{\arctan x}{x}dx}[/tex]
But the term on the left tends to infinity. What can I do? And how do I relate them?
Thank you!