I've been stuck on this question for the last 40 minutes and while it seems fairly straightforward I just can't get anywhere with it and so thought I would ask the wonderful people of TSR for some help!
Also, please excuse me typing it out, I never managed to figure out how to use LATEX.
I need to find the integral from 0 to infinity of exp(-y)cos(xy)dy
As far as i'm aware using integration by parts is a must here so I took the following:
u = exp(-y)
v' = cos(xy)
SO
u' = -exp(-y)
v = [sin(xy)]/x
which gives (using I to stand for the integral from 0 to infinity)
[exp(-y)sin(xy)]/x - I[-exp(-y)sin(xy)]/x
From then I need to use parts again and I take:
u = sin(xy)/x
v' = -exp(-y)/x
SO
u' = cos(xy)
v = exp(-y)/x
Which then gives me:
[exp(-y)sin(xy)]/x - [[exp(-y)sin(xy)]/x^2 - I[cos(xy)exp(-y)/x]]
But from here it all seems to fall apart.
I know what the answer needs to be, I know it has to be:
exp(-y)[xsin(xy)-cos(xy)]
------------------------------
1+x^2
but I have no idea how to get to there.
Thanks very much.