Integral 1/1-x^2 two ways

In this video, I compute an antiderivative of 1/(1-x^2) in two ways: 1) Using partial fractions and 2) Using hyperbolic trig substitution. From those two approaches I derive a really cool identity, which shows that you can derive mathematical identities not only using differentiation, but also using integration. Enjoy!

Note: Credit goes to footskills / Zach Lee for coming up with this idea, thank you so much for the suggestion and the main ideas!

[benjaminkhadem]

I really really really really like this guys energy when it comes to solving this problem. He managed to get me from being miserable/hopeless to attentive/optimistic

[juanfransanchez]

I wish I could enjoy watching as much as you did making it. You're great. Thanks for sharing.

[zanti]

I'll take a box of whatever he had before this video was filmed.

[gnikola]

I came across with this during my calculus classes when trying to solve this integral!! It's interesting because the domain of the ln() version is R - {-1;1}, whereas the domain of the atctgh() version is just (-1;1) (well, depending on how you define the domain of the original function anyways but I refer to the "natural" domain of the expresions if considered as functions). In fact, now that I think about it, the identity you got to, is only valid for when x is in the interval (-1;1). Man I love maths <3

[adamjam]

12:00 when the math makes you progressively and irresistibly more and more insane with every new delightful identity you stumble across xd

[pco]

It's also nice to make the substitution u=ix so you end up with k•arctan(ix)+c

[IoT_]

1:41 when you don't have either complex or repeated roots, pfe could be done in a much simpler way : firstly, you get rid of the first denominator, for instance, (1+x) and put the root of the polynomial (1+x=0, x=-1) to the remaining part : B=1/(1-(-1) =1/2. To find A, we can do the same procedure, namely, 1-x=0, x=1. A=1/(1+(1)) =1/2. This method can help to perform pfe just in one's mind.

[brners]

very smart of you to record from side of the board

[thisismycoolnickname]

I solved it by writing "1+x-x" at the nominator and splitting it into two integrals.

[ZoneEEEEEEEEEEEE]

arctanh(x) exists only in -1<x<1. The identity is true inside that domain.

[shashwat]

Thanks sir, i was having trouble in this integral, now i will never forget it .

[ChefSalad]

Everyone, listen, the inverse of tanh(x) (pronounced tanch, rhymes with branch) is called artanh(x), not arctanh(x). The ar means area. The reason for this is similar to reason that the inverse of (circular) tan(x) is called arctan(x). The input to tangent is the arclength of the section of the unit circle for which you are calculating the tangent (not the angle as you would think, although those things are the same for unit circles, and we are talking about unit circles). This means that arctan(x) function gives the arclength for a given tan(x) value and hence the name "circular arctangent function" or arctan(x). It's similar for hyperbolic functions, except that the input is an area instead of an arclength, and thus, the inverse function returns an area for a corresponding tanh(x) value, and hence the name "hyperbolic areatangent function" or artanh(x).

[ninoskagonzalez]

I love you, you turned something complicated into a very simple thing! Thank you!

[LostAlienOnEarth]

I'll just say that Dr. Peyam must be the best person in the world right now.

[coolcat]

is this a thing?
1/(1 - x^2) = 1/(1 + (ix)^2) .... Integral(dx/(1 + (ix)^2)) = 1/i(arctan(ix)) + c

[janitgjay]

hit my question right on the head, I genuinely appreciate it.

[pascaltchakoute]

"Or should I say... This inverse tanhent". Had me dying for a solid minute. Best maths joke ever, oml. 11:32

[Vandarte_translator]

I definitelty prefer partial fractions.

[jordanvanekelenburg]

I actually did something similar back in highschool. I showed the identity of the arctangent by using complex analysis on a mathematical model (Newton's 2nd Law) of a space rocket back in highschool.

[mertozturk]

heisenberg's son from breaking bad.... it is really interesting that he quit acting and trying to be a math proffesor... proud of you keep doing what youre doing. nice video btw I am sure it will help me to pass my exams