Question

Here's a fun integral. \[\int\limits_{0}^{\infty}e^{-tx}\frac{ sinx }{ x } dx\] for t>=0.

Answer 1

depends on your definition of fun

Answer 2

its an improper integral on both upper and lower limits

Answer 3

http://www.wolframalpha.com/input/?i=integral+from+0+to+infinity+e%5E-xsinx%2Fxdx

Answer 4

Doable. Proof right there. =)

Answer 5

different answers for different t?

Answer 6

Well that's just a special case when t=1.

Answer 7

looks like the answer is arctan(1/t)

Answer 8

i would solve this by expanding the taylor series

Answer 9

this does not have an elementary antiderivative

Answer 10

This has an elementary antiderivative.

Answer 11

Your guess is actually quite close.

Answer 12

ok let me try the taylor