Taming another terrifying integral

Here's an unexpected form for one of our favourite integrals. Incredible how symmetry can simplify such terrifying structures into familiar friends.
The gaussian integral evaluated using Feynman's trick:

[broadecho]

Never thought there would be an integration asmr on YouTube but here we are

[dscasg]

Damn...very smart way to evaluate the Gaussian integral.

[MrWael]

Very smart substitutions. Good job. Thank you very much.

[taterpun]

Before I watch, here is my attempt:

int(0, inf)[x^-lnx•arctanx]dx

substitute u = lnx

= int(-inf, inf)[e^-x^2•arctan(e^x)dx

Break integral into odd and even parts

1/2int(-inf, inf)[e^-x^2(arctan(e^x)+arctan(e^-x))]dx

= 1/2int(-inf, inf)[e^-x^2(arctan(e^x)+arctan(1/e^x))]dx

Use identity arctanx + arctan(1/x) = pi/2

pi/4int(-inf, inf)[e^-x^2]dx

= pi•sqrt(pi)/4

[Santiago_CastellanosG]

Excelent video! i want to ask you a question, Do you do this videos in a phone, or in a tablet?

[emanuellandeholm]

I came up with a nasty integral the other day: ((1 + x^2) arctan x + 1/2 sin(2 x)) sec^2(x) / (1 + x^2) . There is a rather simple primitive function to this mess.

[Ahmed_Magdy]

I just want to be you, I love math so much

[jonsmith]

sup math 505 love your videos
you are integral to my wellbeing

[Ahmed_Magdy]

My name is Ahmed like Ahmed's integral