An interesting variation of the gaussian integral 

Fantastic solution development! You can skip a few steps by invoking Glasser's Master Theorem (or in this case the Cauchy-Stromilch substitution) to jump from your sixth line (reabsorbing the 2 to make the bounds of integration R) all the way to your second to last line.

Very beautiful integral.A constant multiple of the Gaussian integral.

We need more holiday integrals!

Thank you for your innovative video. Hope you enjoy your vacation.

Interesting as always

Good to have you back

generalizing the result, by having e^{-(ax²+b/x²)} one obtains ½√(π/a) e^{-2√(ab)}

Great video! I hope you will have many more successful vacations❤

Welcome back. Glad you had a good trip, Nice problem to solve. My brain had just started to shrink. It's better now. ;-).

at the t substitution, that would be a good idea to prove that x-1/x is monotonic. that's a great video anyway, thanks a lot!

Wow i solved this in class before. Glad i came to the same answer.

Welcome back and Happy Halloween !!

Great Problem, great solution. Thank you. I did it even simpler, with a nice symmetrischen Argument. Let I= 2 J, where in J the integral runs from 0 to oo. The Substitution x-1/x =s gives J=(1/2) Int(-oo,+oo) exp(-s^2)(1+s/(2+s^2)^(1/2)ds =(1/2) Int(-oo,+oo) exp(-s^2)ds +0 (integrans is odd) = sqrt(pi)/2 Hence I=sqrt(pi) q.e.d.

Spotted that factorisation at the beginning straight away because it was on the MAT test today 👀

I'm stupid my first thought was "so... e^1?"

im in 10th grade currently and i started doing integrals in 9th. your videos helped me greatly to improve my understanding of calculus but i have a question. why at 6:56 there appears a 1 in (1+1/x²)dx

Actually there is a formal for that The integral form negative to positive ∞ of f(x) Is equal to the integral of the same limits of f(x-b/x) where b is a non negative number And they both equal the the integral of the same limits of (1/√b)(1/x^2)f((√b)(x-1/x))

we use my homie shlomlich to turn it into a sum of Bessel functions of the first kind of order 0 J0(nx)LOL this was my favorite formula from back when I took complex analysis. edit: I think I'm confused... ahh I'm sure some smart dude can correct me.

Happy halloween!!! I remember that you did the same or similat integral…🤔 I have a nice challange! Nightmare integral 🖤💀🎃: ∫[-1,1]((√((1+x)/(1-x))ln((2x^2+2x+1)/(2x^2-2x+1)))/x)dx The answer is 4π(arccot(√φ))

so beautiful!!

asnwer= oo isit

Nice integral equation just need to learn some integral formulas to apply logics and get the answer

Dude in the video you solved this integral using feynman's technique I prime did not agree with the our solution's derivative as I'(0)=0 but our solution did not equal 0 (since it is - sqrt(pi)) why is that?

A me risulta I=√π/e^2....basta fare il cambio t=1/x e sommare i due integrali 2I=...a fattore abbiamo esattamente la derivata dell'esponente..

Is there a named distribution corresponding to this integral?

❤

hhh i like the small picture