The Internet's Hardest Integral

I solve the indefinite integral of the square root of tan x with respect to x.

[learn-with-vince]

This integral is the go to "hard" integral for most math channels. Usually their solutions are less than quick and dirty though...

[QuillPGall]

learn with vince? nah, lawn with vince

[patricksarama]

Always gotta remember that +c at the end

[NicolasCloutier-ux]

I trust that you can solve, quick and dirty, any integral ! Easy money for you

[ashvanthvijai]

Consider this method
You don't need partial fractions
After you reach 2s²/s⁴ + 1 now split 2s² as s² + s² in the numerator now add and subtract 1 in the numerator
(s² + 1 + s² - 1)/s⁴ + 1 now split the denominators accordingly
(s² + 1/s⁴ + 1) + (s² - 1)/s⁴ + 1
Now this is depressed quadratic by depressed bi quadratic
which is directly solved by u substitution

[TerenceTao-wi]

Now try this insanely hard integral :
Integral between 0 and pi/2 of tan(x)^(x/(e^x)).

[jamelbenahmed]

What about the xth root of tan(x) dx ?

[pagman]

Looked pretty clean until you used partial fraction decomp 😂

[rahoolnarayanan]

hey man theres a much better way

just take tanx= t^2 so you get (secx)^2 dx= 2tdt => dx=2tdt/1+(tanx)^2 = 2tdt/1+t^4

so you get integration (2t^2/1+t^4)dt