A RIDICULOUSLY AWESOME INTEGRAL WITH A BEAUTIFUL RESULT!!!

Here's a ridiculously awesome integral that I'm pretty sure rick Sanchez solved in his head within 2 seconds by the time he turned 3.

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[jkid]

Euler's wonderful *buuuurp* wonderful formula, Morty

[yoav]

Very nice! The one who came up with this integral is so creative!💯

[kappasphere]

Why is it so hard for me to imagine Rick doing actual maths

[emanuellandeholm]

Loved your previous result with the old Euler-Mascheroni and what not. This looks like another spicy boi!

[Kritic_lyr]

Solving along the vid and finally getting the answer pulls out a that'll put 7 year olds to shame lol. Awesome vid thanks !

[SuperSilver]

Yeah I think when you got to step that has e^(e^(2*i*x)), I substituted the definition of sine function in terms of complex exponentials. From there you can make a sub that z = e^(i*x), and then use residue theory on the resulting function. We can treat this result as if we integrated around a closed contour, maybe a circle with r > 1, this should give you 2*pi*e*i, then just find a quarter of this result and take the imaginary part? I think there might need to be more arguments for those last steps, but I think it’s a simple as your argument near the end, you get double the result if you integrate from [-pi/2, pi/2], but because the integrand is also periodic, this interval is the same thing as saying [0, pi], so from [0, 2pi], is just the other half of that result.

Cool problem!

[Saiwastaken]

Hey math505, (unrelated question) which books from springer do u recommend a student who is going to university.... the first topic is infinite sequence and series....

[JirivandenAssem]

Dear sir, I am a highschooler and I have much trouble with understanding When you can and can not switch order of operations in integrals & summations. I have checked wikipedia but the kind of math notation is a bit above my understandings, could you help me out? Or do you know a good site to learn this? Love ur vids, I try to understand everything but this is just getting in the way now :)))

[insouciantFox]

What's wrong with extending the imaginary part argument to csc(x)?

[petterituovinem]

now try to find an integral which result is e times pi times gamma

[sagnikbiswas]

You love Taylor series, don’t you? Haha just messing with you, nice video!

[jonhanon]

When I saw Im [e^(e^(2ix)) e^(ix)] / sin(x), I thought of rewriting it as Im [2 i e^(e^(2ix)) e^(ix)] / [e^(ix) - e^(-ix)] dx = 2 Im [i e^(e^(2ix))] / [1 - e^(-2ix)] dx

As is mentioned in the video, because the original function is even * odd(odd) / odd, the resulting function is an even function.

I then thought to make a substitution of z = e^(2ix), making dz = 2i z dx, or dx = dz / (2i z)

This makes this 2 Im[i (e^z / (1-1/z)) / 2iz]

Applying that this is an even function, we get Im(e^z / (1-z) dz)/2.

This gives us that we can use residue theory to get the at the residue of z = 1 of simply e/2, and therefore, the integral is simply pi*e.