Integral sqrt(x^2+1)

In this video, the second one of the trig integration trilogy (the first one being the square root of 1-x^2), I find an antiderivative of the square root of x^2 + 1 using a trig substitution. Along the way, I also find an antiderivative of sec^3(x). Enjoy!

[DanielSAraujo]

Subscribed right away. You look very happy teaching so I think It makes me less likely to cry whilst studying,

[ThAlEdison]

√(x²+1) is always positive, √(x²+1) > |x| always, therefore even if x is negative √(x²+1) + x is positive.

[artix]

You can solve this using hyperbolic functions by setting x=sinh(u)

[yuvalpaz]

like erick said, it is easier with sinh. generally i think that when you have sqrt with minus you use Trigonometric functions
and for plus inside a root you use hyperbolic functions, it is become very clear why when looking at the definition of those functions.

in the last part: |x|<sqrt(x^2+1)
so
hence sqrt(x^2+1)+x>0

[andresvivas]

Awesome Dr Peyam... Keep doing that awesome job... Greetings from Colombia..

[ThAlEdison]

I think this one is easier if you use x=sinh(t)
x=sinh(t) -> dx=cosh(t)dt -> ∫√(x²+1)dx = ∫cosh²(t)dt = ½∫(cosh(2t)+1)dt = ¼sinh(2t)+½t+C = ½sinh(t)cosh(t)+½t+C = ½x√(x²+1)+½sinh⁻¹(x)+C
Also, using the hyperbolic functions to integrate the upper half of a hyperbola...

[tonypalmeri]

A very special holiday message to fellow math enthusiasts: y=ln(x/m-s*a)/r^2
You have to do some algebra to get the message. Show your work!

[slavinojunepri]

Awesome! I also took notes of the sinh substitution as a cool alternative and the fact that integrating sec(x) gives us ln|sec(x)+tan(x)|. Thank you Dr. Peyam!

[theoleblanc]

Other sub:

√(x^2+1)=x+t
(Euler substitution)

[ajiwibowo]

Awesome Dr. Peyam. I wonder if you teach me hahaha!

[physicsphysics]

Why is there always unexpected parentheses?

[JashanTaggar]

YOO I LOVE YOUR VIDEOS!!!! CAN YOU PLEASE DO THE DERIVATIVE OF f(x) = x!

[wwebadgerse]

i actually managed to do this myself before watching, so proud since i havet even learnt trig substitution

[ryanyork]

How can anyone have the heart to dislike this video?

[milatheresereyes]

SOO grateful for you and this video! I couldn't understand how to solve this problem and this is literally saving me so much time Thanks Dr Peyam!
also 4:36 "integration by parts, don't break my heart" haha

[visual]

Nice video as always Dr Peyam! I have a question: Why does theta have to be between -pi/2 and pi/2?

[poetu]

how about the substitution u=sh(x) (hiperbolic sine)?

[faresberarma]

A tricky way to do it by using x=i*sin(t) (my personnal method)

[yusrafatima]

this video really helped me.. Thank you:)

[NAMEhzj]

Nice video, but did you notice that you write dx = sec(t)² at 3:05 because its the derivative of the tangent, Then you say sec(t)*sec(t)²= sec(t)³ and THEN you pull them apart again and write sec(t)² as the derivative of tan(t) again. Thats kind of silly^^ (well maybe not because you also use the integral later because it comes up on the right hand side..)