A RIDICULOUSLY AWESOME INTEGRAL: int 0 to 1 (x^2cosh(arctan(x)))/sqrt(1+x^2)

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[@vybs9235]

I always hated hyperbolic functions, and now they're in integrals too this is fun 😅

[@Daniel-yc2ur]

Here’s a awesome series as a suggestion: n=1 to ∞ Σ ( F ₙ / 2 ⁿ⁺¹ )
Where F ₙ is the Fibonacci sequence: F ₀ = 0, F ₁ = 1, F ₂ = 1, F ₃ = 2, etc…
The sum evaluates to 1!

[@holyshit922]

Subsitution t = arctan(x)
then integration by parts twice
is the easiest way to calculate it

[@yoav613]

Yes,i also solved it this way

[@MrWael1970]

Smart solution plan. Thank you for your amazing effort.

[@saraandsammyb.9599]

Oh my gosh I love all of these integrals except I don’t understand some of the little manuvers that you go through! I know you explain them but for people who don't know all of that math it’s hard to understand. Like for instance I know up to calculus 2 but I cant understand most of the stuff even though it looks like I should. Because we never learned stuff like the beta function, euler macheroni constant etc.., But I mean could you possibly make a course explaining stuff that isnt so trivial like above the calc 2 level? or do you have any ideas for content I could watch to learn that stuff to be able to understand most of your stuff? I guess what I’m trying to ask is: what level math is this and how do I learn it?! Great video again, thank you for posting!!!

[@henryubah5031]

I was busy counting how many times you wrote *"stuff"* it's exactly 5 times.

[@romanvolotov]

so, in the beginning I wanted to see the graph of the integrand and used desmos. and it looks like a tick, similar to |x|, if you zoom out far enough (it has some curvature around 10 but then straightens out). so I wondered if there's an asymptote but I don't trust my eyeballing and I was unable to find a tool to calculate it for me. in desmos I played around with y=kx+b and found out that when k=2.5 and b=-2.5 (approximately) it looks very much like an asymptote. can you plz comment on this and maybe explore this in a future video?

[@CryToTheAngelz]

For large x, x^2/sqrt(1+x^2) tends to x, arctan(x) approaches pi/2 and cosh(pi/2) equals 2.509

[@romanvolotov]

@@CryToTheAngelz ok, makes sense. forgot about that arctan property. ty

[@sergten]

How does one design an integral? Start with the answer and work you way backwards, throwing in expansions and domain switches along the way? Ever designed an integral for someone to solve?

[@fetch7312]

I'd imagine for a definite integral, you can throw caution to the wind somewhat. with the amount of strategies available for definite integrals its pretty safe to say a lot of definite integrals can be dealt with if you put some thought into the process. Indefinite integrals likely have to be designed more meticulously, as they can be pickier and many functions have nonelementary integrals.

[@maths_505]

Nah starting with the answer is boring. I prefer throwing in stuff while taking care of things like singularities and zeros. Then making sure that the structure converges by looking at a graph or Dirichlet's theorem.

[@sergten]

@@maths_505 Maybe you can make a video of doing that - that'd be ridiculously awesome.

[@giuseppemalaguti435]

Io avevo fatto subito x=tgu, ma poi era troppo complicato... Forse era lo stesso..

[@SuperSilver316]

How this crazy ass integral just gonna have an antiderivative for no reason. Shits wild.

[@user-zb2uh5lk8c]

3:52 Why sec(u) and not |sec(u)| ?

[@maths_505]

Sec(u) is positive on the interval.

[@user-zb2uh5lk8c]

@@maths_505 exactly! I thought that sec u = 1/sin u instead of 1/cos u. my mistake