Question

Compute integral sin x /x from 0 to infinity.

Answer 1

http://www.physicsforums.com/showthread.php?t=64859

Answer 2

doesn't have answer

Answer 3

no but it could help

Answer 4

You have to memorize the solutions to these. Don't try to actually solve it. The answer is just \pi/2\

Answer 5

I know the result. I want to know how :D.

Answer 6

you'll have hard time getting help here for anything above pre-calculus

Answer 7

and elementary statistics :)

Answer 8

If you are currently in cal 1, you probably won't do the series stuff until the end of cal 2

Answer 9

I'm going to upload an attachment soon, just hold on. Integration by parts SUCKS

Answer 10

i think this involves advance calculus

Answer 11

This does involve advanced calculus... this is going to take longer than I thought

Answer 12

can you tell me the idea?

Answer 13

you can use laplace transform

Answer 14

http://answers.yahoo.com/question/index?qid=20080129204110AAnGtsz

Answer 15

This is going to use Taylor Series convergence

Answer 16

Honestly, I don't feel like proving that the Taylor Series converges. That would take about a week by hand for me (because I don't actually know how to do it yet). However, I'm assuming that it converges to pi/2

Answer 17

That's ok. I found it :D.

Answer 18

we can do this using complex analysis:\[\frac{1}{2}\int\limits_{-\infty}^{\infty}\frac{\sin(x)}{x}dx=\frac{1}{2}I \left( \int\limits_{-\infty}^{\infty}\frac{e^{ix}}{x}dx\right)\]
this is the same as:

\[I(\frac{1}{2} \pi i ( res_{0}\frac{e^{ix}}{x}))=I(\frac{\pi i}{2})=\frac{\pi}{2}\]

Answer 19

we need the residue at zero since we have a simple pole there