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Integral of sec(x) but without that trick!

blackpenredpen

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Integral of csc(x) without that trick
Integral of csc(x), without that trick!

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20 Comments

@blackpenredpen Creator 7 years ago Pinned

Red or black?

@pbj4184 5 years ago

Ah this "mechanical" evaluation is much better than the "magical" original because this shows all the relevant motivation and isn't too hard either!

@changenow9228 7 years ago

100 differential equations please...thank you

@johnwalker1058 1 year ago

I really appreciate this, because when other sources show the "standard approach," I was always like "how did they know/think to multiply numerator and denominator by (secx + tanx)?" It feels like you would have to already know the solution beforehand to come up with that, and the rest of the work feels more like verifying an end result that was already found to make sure it works than actually discovering it for the first time.

@GreenMeansGOF 7 years ago

I think it all comes down to already knowing the answer. The second way is more intuitive but the first way is faster.

@Nonsense_2021 7 years ago (edited)

I'm in my first calculus class, and just today I was wondering how to integrate that thing. Coincidence or destiny?
Great vid as always. Greetings from Honduras :D

@itszeen7855 7 years ago

Love it when there is a new upload when I get home from school!

@ishaanivaturi2387 7 years ago

This is so nice! I was curious about the same thing last year and ran through this same derivation too! Glad to know you're bringing it to everyone :)

@CornishMiner 7 years ago

I miss the "Black Pen Red Pen YAY!" intros.

@VibingMath 7 years ago

Thanks man! This technique is also applicable to integrate cosec x and it's more natural to do in this way rather than multiplying a complicated expression in the first place! I will let my students to watch this so they dont need to memorize that trick! Yay!

@akshatahuja2523 7 years ago

RED

@yashovardhandubey5252 7 years ago

This was uploaded at 3 am in the morning in my country...... Still watched it

@Daniel99-tdb 6 years ago

you have a real heart of a teacher, love to see it

@shambosaha9727 7 years ago

This video really made my day

@ieqvilamaria619 7 years ago

Bprp Thanks for yours videos, them are awesome so much, I am your fan!! :)

@JohnSmith-rf1tx 7 years ago

CHEATER!!! :)

You claimed you would do the integration without using the the trick of multiplying by (sec x + tan x)/(sec x + tan x) but that's exactly what you did in the process of converting your answer to the standard form. You may have dressed it up in different clothes, but it's still there! At 2:34, you first multiply by (cos x)/(cos x). Well, that's the same as multiplying top and bottom by (1/cos x). Then, at 8:20, you use (1 + sin x)/(1 + sin x). When you multiply those two together, you get exactly the (sec x + tan x)/(sec x + tan x) you claimed you wouldn't use.