Fastest way to integrate sinx/x from zero to infty

And we've come full cycle with Lobachevsky's wonderful formula. Here's the proof:

[andrewholland]

I cannot say my expertise in calculus is in 505 level, but i am still wondering, why this channel is not way more popular! Rarely i can find channels in which some slightly more difficult problems are shown with almost every step for solving them! Thank you very much!!

[srinitishsrinivasan]

Nice, another possible way is using contour Integration by writing it as e^iz/z and then equating the imaginary part of the above form

[Ben-wvht]

If L(f)_s is the laplace transform of f in s, and L^-1(g)_x is the inverse laplace transform of g,

You can use the identity
Int [0, +oo] f(x)g(x) dx=int[0, +oo] L(f)_x L^-1(g)_x dx

And you also get the result in the blink of an eye

[cytos]

Solving the dirichlet integral in UNDER 2 MINUTES
Step 1: Plug the integral into wolframalpha

[guyzan]

It's the fourier transform evaluated at DC.

[manstuckinabox]

The function obviously representing pi/2* how quick you can solve this integral.

[מיכאלקונטרוביץ]

The only problem with this is that the proof of the formula itself is pretty sophisticated, so I don't consider this way of solving as "fast"

[holyshit]

His name should be read Wobachevsky
(Your W sounds like Russian Л)

[vikraal]

I would have done it with Taylor series but this is far quicker

[nightmareintegral]

Nice integral trick.
You can also use laplace transform indentiti because it is from 0 to infinity or use ramanujan master theorem.

But pls HANKEL!
💔