Integral of sec^2(x), a totally unnecessary way!

Today, we will integral sec^2(x) but NOT using the fact that d/dx(tan(x))=sec^2(x)

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Thank you,
blackpenredpen

[AndDiracisHisProphet]

way to complicated! you could just have used that the derivative of tan(x) is sec^2x! disliked!


just kidding :D

[RamsyC]

My professor did this integral with like 4 or 5 strange substitutions. I guess he was just flexing

[IzayoiMeiHigu]

4:30 the true speed of thinking of an asian mathematician xD
PS: Why on Earth could someone dislike an innocent and helpful video like this?? why??? it doesn't make sense... this is just to improve the world... one of the few channels which actually does an appropriate use of YouTube...

[wellbangok]

I am watching math videos in the summer at 1:00

[mohammediqbal]

When it is simple, we dislike, but when it is uselessly overcomplicated, the the like ratio becomes 100/1
We are just masochists...

[whiz]

Math isn't entertaining unless someone suffers.

[runtime_engineer]

Here is a quick way to know the derivatives of tan(x), csc(x), sec(x), cot(x) - unless you memorize them.

Sec > Sec < Tan
Csc > -Csc < Cot

This was posted by a reddit user MalignedAnus. Derivative of a function is what it is pointing at. Check this out.
Derivative of Sec(x), you may ask, take a look at that table and find what Sec(x) points to - it points to Sec(x) Tan(x) - that's your derivative. It works for rest of them too.

Tan(x)' = Sec^2(x)
Csc(x)' = -Csc(x) Cot(x)
Sec(x)' = Sec(x) Tan(x)
Cot(x)' = -Csc^2(x)


blackpenredpen, maybe you should pin this so others can see :)

[WithASideOfFries]

Isn't it a bit strange that this video has far fewer dislikes than your straightforward approach?

[MathNerd]

I enjoyed the video. Also, using the complex definitions of sine and cosine I found out another anti-derivative of sec^2 (x) is:
-2/(i*(e^(2ix)+1))
I actually got worried about this one because I expected to get the complex formula for tan(x):

Yet when I looked again to see if they differ by only a constant, I got that they differ only by i (or -i if you're subtracting the other way)
Thus, since they only differ by a constant, both of them can be accepted. :)

[dimensionaltravelers]

Thank you! While most people want just the easy way, I always want to know HOW we got the easy way

[MagicGonads]

Upside down integral looks exactly like a proper way around integral?

[Bedoroski]

Integrating (tanx)^2 truly is an eye-opener for me. Not unecessary at all

[daizhao]

I really love integration. But all my friends thought I am bring and crazy. Thanks for your video. Love from China

[morgengabe]

I feel like I'm completely unimaginative when it comes to integration.

[jasonryan]

"This is just the appetizer, and the main course is over here!😅😂"

[deborahfromsa]

You made a mistake in writing the derivative of sinx is -cosx in the by part method, it is cosx

[wontpower]

They probably just disliked the fact that they have to have the integral of (secx)^2 memorized. It might have been helpful to them for you to derive tanx using the limit process.

[preetham]

This is what happens when you first learn integration before differentiation

[Steveiegaming]

please do a video on indef integral of: sqrt tan(x) dx please. Nice videos btw! Keep it up!

[holyshit]

Rewrite with cosines and sines and then pythagorean identity + integration by parts