Why we dont need the absolute value when we do trig sub https://youtu.be/KzfQ49JkxzE?si=IX6BVQXi5qvUumBe

Wow, I had completely forgotten this manipulation. But it’s been 40 years! Nice video as always.

@9:52 Your body language speaks volumes about your opinion about those kinds of teachers… ;)

Cant believe i used to know how to do this a year ago. Less than a year of no math and i already forgot everything lol

I couldnt find any math online worth doing and then found this (sub?) channel i literally just scrolled down and saw the next class im gonna take when i go to college in a 55 video playlist by you thank you so much i hope i can teach like you teach someday!

followed advice, huge help. THANKS!

I love this channel

You can also do u sub but needs to factorize the numerator
(√(1 - (1/x²)))/x³
Then do u sub inside the root

Hi bro! Just wanted to say that you're doing amazing work!

A fellow math enthusiast here! I love solving those tricky problems too!

Keep the great work going!

Masterclass ❤

An elegant, educational, and fun video for learning how to solve integrals. Thank you.

Yes. Yes. We'll done.

This is also possible and very fun with hyperbolic-trig-sub:

∫ ( √(x²-1) / x⁴ ) dx
note: cosh(u)²-1 = sinh(u)²
let: x = cosh(u)
-> dx = sinh(u)du
= ∫ ( sinh(u)² / cosh(u)⁴ ) du
= ∫ ( tanh(u)² * sech(u)² ) du
let v = tanh(u)
-> dv = sec(u)²du
=  ∫ (v²) dv
= ⅓v³ + C
= ⅓tanh(u)³
note: cosh(u)² = x²
-> sinh(u)² = x²-1
-> tanh(u)² = (x²-1) / x²
-> tanh(u) = √(x²-1) / x
== ⅓(√(x²-1) / x)³ + C

Excellent explanation! Thanks.

You an also skip the Tan and Sec usage to just Sin/Cos and 1/Cos resulting in I[ Sin(Theta) *  Cos^3(Theta) * Sin(Theta) / Cos^2(Theta) * d(Theta) ].

Sin(Theta) = sqrt(x^2 - 1) / x
x = 1 / Cos(Theta)

At 8:24 I said to myself, "And du is cosθ dθ, so the whole thing is just u^2 du," and I got a happy little tingle in my brain. 😋

For fun do it with trig sub and without it (fe with integration by parts only)

I tell people to put the "+ C" immediately after the last integral sign goes away. That way, it's absolutely correct. Besides, carrying it thru from that point on, you're MUCH less likely to forget it, as having to remember to add it at the end. JMO.

Also, as you draw the triangle anyway to get the 3rd or missing side... I tell people to draw the triangle RIGHT AWAY, then you don't have to remember which trig identity you need in the integral; it can be read DIRECTLY OFF THE TRIANGLE!!  Less to remember is, to me, ALWAYS BETTER (My memory sux; so this is my workaround so I don't have to remember very much 😊😊 ).

Love your channel, helps me review. In this case can’t we just take out minus sign and introduce x=cos theta?

aye bprp. I watch your channel and think your a good teacher. Could I get your help with a question: I want to have a fishing rod that can cast 100m I want to be able to find the height of the parabola that tells me how high I have to throw it in the sky to fly 100m but I also want the angle of the parabola on the 2nd quadrant of the catesian plane to find out what angle I should release. The second part of the question was what would the new equation variable look like if I had a 5'6 and a 6' rod and the dialation of the parabola. And the third part was throwing 100m throwing the line out at a 120 degree angle from originI also dont know a 3d cartesian plane works at all. Can you please solve this for me boss?
     cheers