Question

find the indefinite integral [(sin(pi*x))/x^2]

Answer 1

\[\int\limits \frac{ \sin(\pi*x) }{ x^2 }dx\]

Answer 2

It seems somewhat unsatisfying to say "look at a table of integrals" but that might be the least painful answer.
\[\int\frac{\sin ax}{x^n} \mathrm{d}x = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}} \mathrm{d}x\,\!\]
You could probably find that yourself via integration by parts.
The second part, though, will end up like this:
\[\int\frac{\cos ax}{x} \mathrm{d}x = \ln|ax|+\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+C\,\!\]
And I'm not at all sure how to begin explaining how that works. :D

Answer 3

As crazy as this will sound it somehow seems to equal zero or maybe I am not understanding this. I used Maple to get the right answer but I don't understand maybe you can follow what happened on this attachment

Answer 4

I can sort of follow it, but not really. It is a bit above my level of calculus. If you got the right answer, though... good work? :)

Answer 5

Oh, I see. The thing you're actually wanting to integrate ends up being something with cosine plus the ugly sin(pi*x)/x^2 integral, but then you have something else that subtracts that integral... so you never have to actually evaluate it. :D

I think!

Answer 6

I think too. I use maple to check my work but I don't like when I can't do it by hand.

Answer 7

Well, now that you know that awful thing ends up canceling, you can probably do the rest by hand and just leave it unevaluated until you get to get rid of it. :D