Integral of tan(x) & sec(x)

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0:00 integral of tan(x)
0:41 vs. integral of sec(x)

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bprp #fast

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When you need to prepare for your finals in 2 minutes

Adomas_B

I really enjoy these fast videos. Keep 'em coming!

PixelSergey

The multiplying of sec x in the integral of tan x might seem outright magical but BPRP is actually being a great magician here by diverting our attention away from the reasoning behind doing that to make it look more isolated and therefore its origins more mysterious.

When we write tan x as sin x/cosx, we notice a great property to be exploited here which is the cyclicity of the derivatives of sin and cos (barring some multiplicative constants which can always be taken out of the integral) So since we have a cos in the denominator, we must have a -sin x in the numerator so that the -sin x dx is absorbed into du when u=cos x and we are left with du/u which gives us ln(u). But since we're missing a -ve, our integral is -ln(u) which is -ln(cos x)

Now if we take the derivative of -ln(cos x), we notice it is -1*1/cos x * -sinx by the chain rule which is sinx/cosx and therefore tanx

But if we apply log rules and take the -ve inside, we see the integral is ln(1/cos x) which is ln(secx). The derivative of _this_ is 1/secx * sec(x)tan(x) and the secs cancel out to leave us with tan. Now we see why multiplying sec x on the numerator and the denominator helps solve this integral. But without the proper motivation, this seems magical and elusive which seems to be intentional here

pbj

Next time with Weierstrass sub (u=tg x/2)😂

SimsHacks

He speaks quite clearly for going fast on those problems.

bettkitty

Asalamualikum.
Can you tell when do we know exacrly that we have to use U and substitute. Anyway to know when to use this technique?

MathswithMoeed

i tried to integrate ∫secx dx, but ended up with ∫1/dx

ramazantaikocusu

I’ve seen it a million times, but that sec(x)+tan(x) trick has never meant anything to me. I personally see it as little more than a meaningless and kind of bizarre trick. Not that it isn’t useful- au contraire, it’s very useful- it just seems to come from thin air.

jessehammer

Dude, the delay with your voice keeps getting more and more delayed :/

agabe_

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