Calculus - The Most Difficult Integral - sqrt(tan(x)) (Request)

**MISTAKE**
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I miscopied the signs on the last term the final answer when I combined both parts of the integral. The computations of the two integrals is correct, so please refer to those! The final term should go: +,- on numerator and +,+ on denominator. Annotations have been added to show these corrections.
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This is the most difficult integral that I have solved so far. If you can think of harder one, let me know! Remember to submit any questions / requests that you may have to get your own solution video like this one! Enjoy.

[TheLazyEngineer]

Thanks for watching, and for those of you that are subscribing, what would you like to see more of? Let me know!

[Eichro]

You just put a tiny square root there and all hell breaks loose

Gotta love calculus

[skoockum]

I saw this problem at breakfast on the back of a box of Capn Crunch ...didn't get a chance to tackle it tho cuz I got stuck on the maze.

[jakobygames]

*Sees intg(sqrt(tan(x))) on paper*
Me: Looks simple enough.
Morgan Freeman: Little did he know just how wrong he was.

[NorthernWindNut]

I can see why engineers like their integral tables lol.

[furkanozdemir]

imagine doing all these work and forgetting + c at the end.

[andyan]

i got a little antsy waiting for the final "+ C"

[mapleace]

So when does this video come out in English?

[guitarttimman]

We had to do this way back in Calculus 164 when I was attending Purdue University. I actually had the entire table of integrals memorized. You're essentially utilizing integration by substitution several times to get to the final answer. I think you do have a good understanding of the subject, and your teaching strategy is very good. I like how you provided a geometric interpretation for Tan x. That is, how you set the side opposite the angle, x, equal to u^2 and the side adjacent equal to 1. You did an awesome job Sir, and think you for bringing back the good memories.

[swankykoala]

Glad I got recommended this 1 in the morning. Was really rewarding to barely understand it with the knowledge I have!

[trazwaggon]

i study ancient languages
what am i doing here

[cankostoev]

Amaizing MAN!!!!Thank you! I am so inspired that I am watching it over and over....

[alkankondo]

The ... "Lazy" Engineer?!? On the contrary! This was an excellent example using so many integration techniques - and well-explained, I must add!

[perkodanny]

This is a great problem, and he did a great job explaining the solution. But, did it bother anyone else that he draws his integral signs from bottom to top? That's just wrong.

[voodoo_child]

Finish integral of 2u^2/(1+u^4) with partial fractions.

[eddiesrbn]

sick so if my test asks me for the deriviative of 1/sqrt2 * + 1/sqrt8 * ln((sqrttanx + 1/sqrttanx - sqrt2)/(sqrt tanx + 1/sqrttanx + sqrt2)) i'll easily be able to say it's the square root of tanx.

[JoeMama-wmgl]

YES....FINALLY, THANKS TO THIS, I CAN MAKE MY REDSTONE TRAPDOOR

[josephkitchen]

I’m a 4th year Biomedical Engineering major. Took a ton of math and this just blew my mind 🤯. I miss math 😭..

[pacolibre]

Commenter: "I've seen much more difficult integrals than this."
Responder: "Oooo like what?"
Commenter: "What? I didn't say anything..."

[guilhermegoncalves]

Denominator: u^4 + 1 = u^4 + 2u^2 - 2u^2 + 1 = (u^2 + 1)^2 - 2u^2 = (u^2 - sqrt(2)*u + 1)*(u^2 + sqrt(2)*u + 1). Then, complete the squares or expand in partial fractions.