basic integration

[mint shuttle]

i didn't understand the proof or integration of (cot, sec ,csc) integrations

[midnight temple]

What exactly don’t you understand about the integrals in question?

[mint shuttle]

the proof of it self
i can solve but feel has small feeling of not understanding why these trig iden like these cot is normal sec and csc like isn't

[cobalt field]

well the integration is 'proved' if you know the derivatives of the trig functions (and a few other functions)

[midnight temple]

It depends what you mean by a proof.
A proof would be to check that the derivative of the anti-derivative you get is indeed cot(x), sec(x) or csc(x). This shouldn't be too hard.

The integration methods for those all use some form of trig identity with substitution. If you can point out which of those exactly you're having trouble with, then it may help. Otherwise it's hard to say what you're having issues with really.

Like is it more that you don't understand the identities used, or that you don't get some trick that was applied, or that you don't get the substitution that was used, etc.

[vale pecan]

Are there different meanings to what constitutes a Proof in Mathematics? Different types sure. But a proof is a proof. Only thing even remotely contentious is reductio ad absurdum.

[vale pecan]

Show us your proofs and point out where exactly you get stuck.

[midnight temple]

Well the thing is that I wouldn't call integration techniques a "proof," just like I wouldn't call differentiation techniques a proof of derivatives.
It's unclear what they are calling a proof from the get-go.

If anything a convincing proof that ln|sec(x) + tan(x)| + C is the indefinite integral of sec(x) is just differentiating ln|sec(x) + tan(x)| + C and seeing that you get sec(x).

But they don't seem to be meaning that

[vale pecan]

The "techniques" themselves are theorems that have proof though.

[midnight temple]

Again my point is that it's not clear what they are calling a proof. It's not common to say that the computation of an indefinite integral is a proof.

[vale pecan]

Oh. I didn't catch that last bit lol.

[midnight temple]

I suspect that it's probably some trig identity / rewriting trick that is used in those computations that they're having trouble with.

We'll see I guess

[mint shuttle]

yeah i meant the technique to slove it (ik it's already memorised normal one)
and yeah if we took a d/dx of ln|secx + tanx| it got secx fr
but if we took the integral as a normal if we don't know the ln|secx + tanx| function yet
i saw that we it's normal like multiplying the secx with (secx + tanx / secx + tanx) which's one that's normal but heck yeah the first idea got is secx + tanx function like and if they r another methods to make it click?

[midnight temple]

Unfortunately those kinds of tricks (multiplying by 1 in a useful manner) happen a lot and apart from the fact that they simplify things a whole lot, there isn't a nice answer as to how you'd come up with it other than that it works.
There may be other ways though. You could for instance rewrite sec(x) as 1/cos(x) and multiply by cos(x) / cos(x) to get something nice.

[mint shuttle]

and for ∫cscx dx
we use the same way but with (cscx - cotx/cscx - cotx)
but the same idea doesn't click
but the proof that if i diff the function ik it
to check my answer

[mint shuttle]

ik it's fr usefull at many situation and it make sense just in trig with me like memorised thing not truly understood or smth

[midnight temple]

However, note that it's not super common for those types of problems to be found in exercises in calculus textbooks. In general they're left as examples specifically because at this point students haven't necessarily been acquainted with those kinds of tricks and it's hard to come up with them if you've not seen it work in the past.

[mint shuttle]

idk theyr mostly side thing or even non in the calculus excercies but somedays feels like unsteady in trigs integ so just curios abt it feeling if this doesn't make sense so it's familiar not understood or smth
(just in trigs) i solve with the integrations techinques in others normally so that's why when i got stucked in those i asked here if they r can make sense with other method or way

[midnight temple]

Yeah I think it's best not to overthink it. There isn't really an nice explanation to it other than it helps.

And if you prefer other methods there's the one I mentioned above which may feel slightly less "out-of-the-blue" than multiplying by (sec(x) + tan(x)) / (sec(x) + tan(x))

[mint shuttle]

i think the same :-:

[mint shuttle]

the which one?

[midnight temple]

You could for instance rewrite sec(x) as 1/cos(x) and multiply by cos(x) / cos(x) to get something nice.

This