antidiff cot^2?

[Phantom-II]

How do you integrate cot^2? i came across it today and couldnt solve it, the solutions didnt really make much sense either

[pi]

Hint: cot^2(x) = csc^2(x) - 1

[Phantom-II]

yea i got there, but then what do? actually i think my original question might have been int cosec^2, but i didnt go far from these two forms

[pi]

I'm not sure of a proof (you could jump back to the fundamental theorem of calc to prove the derivative and hence the integral if you'd like though), but we know that d/dx(cot(x)) = -csc^2(x), hence, the integral of csc^2(x) in relation to x equals -cot(x) + C

[Phantom-II]

hmm, yeah thats the formula they used in solutions, but theres no mention of it in the essentials book nor the formula sheet, thought it was abit dodgy.

thanks anyways though, guess will have to rmb it by heart

[paulsterio]

I don't think you need to know that the integral of cosec^2(x) is -cot(x), but I'll try to think of an elementary way to do it.

[Phantom-II]

i think the book did sec^2/tan^2 as an alternative,  from cosec^2, which, i thought, was somewhat humorously tedious considering forming that from repeating the basic sin and cos requires 2 fractions within 1

[paulsterio]

i think the book did sec^2/tan^2 as an alternative,  from cosec^2, which, i thought, was somewhat humorously tedious considering forming that from repeating the basic sin and cos requires 2 fractions within 1

OMG I CAN'T BELIEVE I DIDN'T SEE THAT! D: that's the right method

let u = tan(x) du/dx = sec^2(x) then just integrate 1/u^2 and then sub back in

[|ll|lll|]

I don't think you need to know that the integral of cosec^2(x) is -cot(x), but I'll try to think of an elementary way to do it.

Don't you need a +C there even though its an intermediary step?
Just wondering 

[paulsterio]

Yes you would need to add a constant, I would add A and then later on add B

Then I would say let C = A + B, then replace them with C

I was just lazy

Though mathematically you could argue that the constant is maintained within the integral that you haven't found yet, if you know what I mean, but for the trouble that it's worth (even though it might be mathematically correct), I'd go the safe way.

[Phantom-II]

lol, the C
must not write C when it asks for "an integral/solution", is this correct? because i mean, even if you leave C in, it is still an integral with C being any value. And if its because C can be anything, then i am free to write something like +100 at the end of an integration?

[pi]

lol, the C
must not write C when it asks for "an integral/solution", is this correct? because i mean, even if you leave C in, it is still an integral with C being any value. And if its because C can be anything, then i am free to write something like +100 at the end of an integration?

You can write +100 if you want Just don't write +c or +(any letter)

[paulsterio]

You can write any letter you want, because it is an arbitrary constant.

[pi]

You can write any letter you want, because it is an arbitrary constant.

But that's if it asks for the anti-derivative, if it asks for AN integral or A solution, no constants (right?).

[Jenny_2108]

I saw in A+ notes, it said "an antiderivative" => no need to write + c