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I came across with the following type of integration with singularity.

$$\int_{s_2=0}^{s_2=\infty}\int_{s_1=0}^{s_1=s_2}\left(\frac{1}{s_2-s_1}\right)^{3/2} \,ds_1\,ds_2 \, .$$ How can I solve it?

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    The inner integral is already $+\infty$, thus the whole integral is $+\infty$. – 2017-01-10
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    Your only choice is to introduce an artificial cut-off and even then you are only just delaying the inevitable divergence. – 2017-01-10
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    @ Triatticus can you provide some relevant article where such type of divergent problem is dealt with? – 2017-01-10
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    Cross-posted from [physics.se]: http://physics.stackexchange.com/q/304076/44126 – 2017-01-10

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This seems like a straightforward application of the usual rules, applying the power rule gives

$$\int_0^{\infty}{1\over 2}(s_2-s_1)^{-1/2}\bigg|_{s_1=0}^{s_1=s_2}\,ds_2$$

which of course is divergent.

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    yes if we do the simple integration it is divergent. But I'm looking for a technique where I can avoid the divergence. May be I can not get the accurate result but some approximate result would be fine. – 2017-01-10
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    @kg2015 there is no way to "avoid" it, it's what it is. No technique can change water into wine, not for a positive function. If you had one that changed sign, sometimes there are weaselly tricks, but in this case there's no hope, anything you would call "integration" will give the value $+\infty$. – 2017-01-10
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    @ Adam Hughes what I actually mean by avoiding is that if I take the limit of 's1' from '0' to 's2-e(some small value)' can I avoid the divergence. Does there such type of technique in mathematics that do my job ? – 2017-01-10
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    @kg2015 no there is no hope, By continuity of the integral the smaller you take $\epsilon$ the closer you will get to $+\infty$. Yes you can pick **some** $\epsilon>0$ and all will give finite values, but then you won't have anything to do with the original integral, as I can pick a smaller one and get a radically different answer, billions, trillions of times larger than whatever yours is no matter what $\epsilon$ you pick. – 2017-01-10