How to approach/solve this integral?

Question

Could somebody suggest how to approach or solve this integral: $$ \int_{0}^\infty e^{-a t}{2+t-2\sqrt{1+t}\over t^2}{\rm d\,}t, $$ where $a>0$ ? It is not a homework. I tried to use residuum calculation but did not find a path encircling the pole of order 2 at zero that would became the real integral I need to calculate. Other option was to get rid of the double pole to become just a simple pole to be able to indent it (if necessary). So I integrated per partes using $$ v'=1/t^2 $$ and $$ u=e^{-a t}(2+t-2\sqrt{1+t}) $$

I indeed got a slightly different integral with just a simple pole at zero and another pole at -1 (and a branch cut): $$ \int_{0}^\infty {e^{-a t}\over t}\left(-a(2+t-2\sqrt{1+t})+{\sqrt{1+t}-1\over\sqrt{1+t}}\right){\rm d\,}t. $$ This is when I got stucked since I need to get a real integral from 0 to $\infty$ but the branch cut starts at -1 so my indentation plan failed.

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

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Here is a start. Differentiating twice with respect to $a$ gives

$$ \int_{0}^\infty e^{-a t}(2+t-2\sqrt{1+t}){\rm d\,}t.$$

Now, work out the integral and follow the technique used in this problem.

edited Apr 13 '17 at 12:21

Community

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answered Nov 07 '13 at 23:13

Mhenni Benghorbal

47,431

Is it not just parts/substitution from here? – Bennett Gardiner Nov 07 '13 at 23:40
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The Laplace transform can be done and yes, per partes follows and we got the result. Hats off to Mhenni Benghorbal for this elegant method. Almost feel ashamed that I did not find it by myself. – leprotic Nov 08 '13 at 00:34
@leprotic: You are very welcome. – Mhenni Benghorbal Nov 08 '13 at 02:21

How to approach/solve this integral?

1 Answers1