A few improper integral

\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty }{\frac{{{x}^{s-1}}}{1+x}\text{d}x} \\ \end{align*}

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What about them? Do you have a question? –  JavaMan Jan 4 '13 at 4:44
There must be something wrong with MSE at the moment. Your question mark doesn't appear on my screen. –  Michael Albanese Jan 4 '13 at 4:44
What class are these questions from? Those are rather non-trivial for introductory Calculus. –  user7530 Jan 4 '13 at 4:48
You can use the following technique to evaluate them –  Mhenni Benghorbal Jan 4 '13 at 4:49
Do you want to compute them or determine under what conditions they converge? –  mrf Jan 4 '13 at 20:58

1 Answer

All of them can be solved using the beta function technique. See (I), (II), (III). Notice that, the second integral can be written as,

$$\int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x}=2\int_{0 }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x}.$$

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I don't feel like that down votes in previous question you've got. Sometimes, some of us forget that we are here just to share our thoughts for solving a problem and help someone to understand a problem but not to war. +1 –  Babak S. Jan 4 '13 at 6:58
@BabakSorouh: As you see, I just suggested an idea for solving the problem, but I do not know why they made all of this argument!! Thanks for comment. –  Mhenni Benghorbal Jan 4 '13 at 7:03
I made some + for other your answers you made. ;-) –  Babak S. Jan 4 '13 at 7:09
@BabakSorouh: Thank you. I really appreciate it. This website serves a good purpose for humanity. –  Mhenni Benghorbal Jan 4 '13 at 7:11
@Babak Did you actually ma(k)e some + for other (Mhenni's) answers (Mhenni) made because (you) don't feel like that down votes in previous question (Mhenni's) got? Everybody is entitled to one's own opinions about upvotes and downvotes on the site, of course, but if you did vote on some answers by compensation, you might wish to reconsider and to stop doing it: this is misleading other users about your opinion of the posts you upvote, and definitely not the way the system is supposed to work. –  Did Jan 4 '13 at 14:56