Integral of sqrt(1+tan(x))

Integral of sqrt(1+tan(x)), harder than the integral of sqrt(tan(x)),

[LS-Moto]

I feel so honored :) ... Hello to everyone and warm greetings from Belgium 😀😀😀

[aLumpOfParticles]

when you changed the W to the X you wrote + instead of -
btw check the answer via derivative :D

[VibingMath]

What a show man! And also big congrats to Lars for his winning the cancer finally! You two are awesome integral-fighter and cancer-fighter respectively 😎

[meedonexus]

I got A+ in calculus 21 years ago
Integration was my favorite game but after these years I totally lost my skill

I love math more than any other science but unfortunately I left studying it to have better job in engineering to gain more money

Math is the science that all inventions based on it

Love math

[alhassanelkossei]

Please check the answer by differentiation.

[ayoubachak]

there is a misstake sir !!!
when you were substituting u in the w expression
you puted a (+) insted of a (-)
I love your work <3

[seb_]

Do the proof that sqrt(2) is irrational in under sqrt(2) minutes!

[ernestschoenmakers]

The other method is partial fraction decomposition where you divide u^4-2u^2+2 by
u^2-au+b. After working this out you'll get: a=sqrt(2+2sqrt(2)) and b=sqrt(2).

[tomatrix]

Those who are womdering why he used hyperbolic cot instead of hyperbolic tan, it is because with tan there is a limitation, namely |x| <1 but the limitation of of coth is |x|>1 so that limitation works here

[leponpon]

Keep making more of these amazing videos! The world needs more of this!

[wuyqrbt]

I love the hidden symmetry in this integral!
Great ... I just enjoyed.
Thank you so much ❣️

[maskedman]

youtube must encourage these typeof educative channels

[QuantumHistorian]

Way easier to start off with the substitution 1 + tan(x) = cos^2(u) and then do some simple trig until you can integrate by partial fraction. Gives a MUCH nicer answers too: 1/sqrt(2) ln[sqrt(2) + sqrt(1+tan(x)) / sqrt(2) - sqrt(1+tan(x))] + c

[masonholcombe]

when you went from u to x when integrating, you did + instead of - for u! all good though, nice job!!:)

[JulesvanPhil]

Very nice video :-) But you did a mistake in the last line: when resubstituting the w you wrote a plus instead of a minus :D

[edusoto]

The polynomial P = u^4 + 2 u^2 + 2 factors over the reals (the only irreducible polynomials over R are linear or quadratic with negative discriminant).
Here is a factorization
P = (x^2 - a x + b) ( x^2 +a x + b)

where a = sqrt(2sqrt 2 + 2) and b = sqrt 2

Once you compute this, the integral is straightforward.

[nchoosekmath]

Wow, this one does not even involve special function in the answer. But the steps were really long. Nice video!

[AmooBaktash]

A sign error happened at 16:34! Note that w = u - sqrt(2)/u.

[rashmigupta]

Your change of face expression at 2:41
😂😂

[Mernusify]

You could also write the answer with tanh^(-1). The differentiation for this is MONSTROUS (and fairly tedious), but it's doable.