integral of e^tan(x)

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How do we do the integral of e^tan(x)? This integral looks very impossible with our typical calculus 2 integration techniques. Here we will need non-elementary functions Ei(x) and li(x). I will show you how to get the answer WolframAlpha gives for the integral of e^tan(x). Note: the result should be real despite how non-real it looks since e^tan(x) is real, to begin with.

@blackpenredpen

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This is the answer on WolframAlpha. It doesn’t look real but it should be since the original integral is real. I apologize for accusing the answer to be non real. Thanks for the people who pointed it out.

blackpenredpen

the fact that you get really into the math that you work on in these videos and like learning more about it immediately makes you one of my favorite teachers on the subject. I'm slowly picking up bits and pieces of calculus, but I'm able to keep up and understand the content you show, and I really enjoy learning these tricks from your channel here! It's encouraging me for my upcoming calculus classes next year, so thank you for making higher maths interesting for me!

takeoverurmemes

Whoa!! Thank you bprp! This made me thinking if there will be a series of videos to integrate e^trig (from sin x to cot x), this will be so satisfying!

VibingMath

I gather that the real parts of the expression inside the parentheses cancel out leaving the imaginary parts to interact with the i outside, leaving a real expression.

iambic-kilometer

You deserve at least 3 Million Subscribers! I don’t know why people don’t like math and ignore your videos. 😔

calcoholic

Subscribe and you will get more cool integrals!

blackpenredpen

You know you're early when there's no comments

JuniorTheCuber

Cool! So I guess it would be a little more tricky if you want to do e^sin(x) as derivative of arcsin has square root. Doing partial fraction is probably not an option??

nchoosekmath

Please tell us more about non integrable functions
Why is functions non integrable ?
Is it because we haven't find out enough techniques to integrate function
Or
Is it because the integration of certain functions do not

ishworshrestha

“Check this out!”😂

I love his passion... You’re a fantastic teacher 👌👌

samuelatienzo

why are some functions like non inegrable, if we just make up like random functioons like error functions? if e^tanx is integrable with non elemntary, but e^sinx and e^cosx is not intebrgable, can't we just make up random functions,

pixelninja

Idea for next video: manipulate algebraically to check if the answer is really complex for x is real or just looking complex but once algebraically manipulated turns out real. Why I question this: e^tanx is real for all x real so the anti derivative should be not have a different imaginary component for different real x cause otherwise the complex wouldnt be in the +c

helloitsme

It seems that at some point mathematics becomes some sort of a sense, and you develop an intuition for equations, even for the most complex ones. I cannot otherwise explain how several people, like Maxwell and Boltzmann for example, developed such complex mathematical formulae to describe the motion of particles and the behaviour of electricity in specific frames of reference.

Mathematics is indeed the way to figure out everything. It.s a gift to us from the universe.

chrisryan

I tried to watch the video about Ei(x) and li(x), but YouTube says it's private :-(

PhilBoswell

The idea of e^tan(x) comes up when someone is trying to brute force a complex rotation matrix (which both scales and rotates a set of complex coordinates) that yields the highest correlation between the rotated domain and the rotated range of a complex least squares regression (often bicomplex number coordinates as well.).

Let assume two vector coordinates: x=(a+bi) and y=(c+di). We notice that is we apply the rotation matrix x_2=xcos(theta)-ysin(theta), y_2=xsin(theta)+ycos(theta), that there exists a very strong quadratic regression from x_2 onto y_2 with an R^2> 0.999, yet the original inputs, x and y, have no significant correlation.

The angle of theta is a complex number, meaning it has both linear (scaling) motion and lateral (rotational) effects on the original x and y vector coordinates (see Aram Boyaijiam's disseration titled "The Physical Interpretation of Complex Angles."

Now suppose that we don't know what theta is, or even if such a theta exists? How would we write an algorithm that tests every possible real number magnitude (the scaling effect of theta) and every possible rotational angle?

We would put our theta in the form of: Theta = e^(tan(z)+iw), restricting both z and w from -pi/2 to +pi/2 (since complex rotation matrices that bring us to the second or third quadrants will produce identical R^2 returns on the regression of y_2 from x_2, we do not need to search beyond those bounds).

Since tan(z) ranges from negative infinity to positive infinity under those conditions, and e^tan(z) therefore ranges from 0 to infinity, we can now write an algorithm that incrementally improves upon its guess of the best possible complex value of theta.

Now for the the strictly two-dimensional case of x and y, a scaling of e^tan(z) won't affect the R^2 value, since Least Squares Regression is invariant in respect to a scaling of the entire data set.

However this changes once we have three or more complex number coordinates, because the best complex angles of rotation are not required to have the same real number magnitude.

Anyway, that's what brought me here.

edwardsolomon