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1) What is the error in the following calculation ?

$\int_{0}^{oo} \frac {sin(px)}{x}dx$=$\frac {\pi}{2}$

derivating by p at both sides

$\int_{0}^{oo} cos(px)dx$=0

But the second integral does not converge.

2) In wikipedia, it is shown how the following three integrals can be calculated with the method of contour integrals. Can they also be calculated with the methods of parametric integrals ?

$ \int_{0}^{oo} \frac {log(x)}{(x^2+1)^2} dx $

$ \int_{-oo}^{oo} \frac {cos(tx)}{x^2+1} dx $

$ \int_{0}^{3} \frac {x^\frac {3}{4} (3-x)^\frac {1}{4}} {5-x} dx $

3) a bit off topic, but can the integral

$ \int_ \frac {1}{(x^2+1)^2} $

be calculated by the method of integration by parts ? I was only able to do this after the subtitution x=tan(t), which gives the integral of $cos^2$(t) , but even to calculate this, I need the additional fact that $cos^2x$+$sin^2x$=1. Have I overseen anything ?

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Generally speaking, you can't differentiate inside of the integral if the integral doesn't converge absolutely. –  Random Variable Oct 31 '13 at 11:12
    
Is the absolute convergence also a sufficient condition ? –  Peter Oct 31 '13 at 16:56
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On page 13 of the following paper, sufficient but not necessary conditions are given. math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf –  Random Variable Oct 31 '13 at 21:23

1 Answer 1

up vote 1 down vote accepted

About point 3), your result is correct. Do you remember what is Cos[2 t] ? I am sure that you can now find the change of variable for the last integral.

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With this, I even do not need integration by parts. Thanks! –  Peter Oct 31 '13 at 16:54
    
And a change of variables is not necessary either. –  Peter Oct 31 '13 at 16:59

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