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 ?
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.