Integral of cot(x)^2 dx

1. Apr 17, 2004

algebrist

well, can somebody please tell me how do i get that:
Integral of cot(x)^2 dx == -x - cot(x)

what technique of integration should be used here?

2. Apr 17, 2004

matt grime

try differentiating -x-cot(x) and see what you get. but if you don't like that remember 1-sin^2=cos^2

Last edited: Apr 17, 2004
3. Apr 17, 2004

algebrist

hmm

well, of course i get cot(x)^2 when i differentiate......
the question is how do i solve the integral without knowing that it's equal to -x - cot(x)

if i use cos(x)^2 + sin(x)^2 == 1 i get to solve
Integral of 1/sin(x)^2 dx
which i don't find easier

4. Apr 17, 2004

matt grime

you know your trig derivatives? that's how you solve these, you konw what differentiates to give what you want, and cot's derivative is cosec^2 or whatever.

5. Apr 17, 2004

algebrist

why should i care about cot's derivative? i don't see your point

6. Apr 17, 2004

matt grime

err, because if F is such that F' =f, then the integral of f is F+k some constant k? fundamental theorem of calc? the same reason why the first suggestion of differentiating -x-cot was offered? if you know the derivatives of basic functions it might help you solve the questions that get set in the exams? any of those reasons interesting? most integrals can't be solved in any nice way, so it's good to know the ones that can.

7. Apr 17, 2004

algebrist

ha-ha-ha do you imply that i just have to know the result by heart???

what i am asking is how to solve
Integral of cot(x)^2 dx

i know the result, i want to know how to obtain it

i have tried integration by parts and trigonometric identities but nothing seems to work

8. Apr 17, 2004

matt grime

the way you solve it is by knowing the antiderivative of cosec^2, because it is an elementary function. If you don't want to learn these things, fine, but don't get shirty.

9. Apr 17, 2004

algebrist

ok the antiderivative of cosec^2 = -cot
so how do i use it?

10. Apr 17, 2004

algebrist

ahhh i see.......

11. Apr 17, 2004

algebrist

but i would like to know how this antiderivative is obtained

12. Apr 17, 2004

algebrist

as i told you i don't find
Integral of 1/sin(x)^2 dx
an easy one

13. Apr 17, 2004

matt grime

Go back to the third post. You did the subs 1-sin^2=cos^2 and you yourself said that you needed to integrate 1/sin^2, so go from there. I'm at a slight loss as to how you would integrate cos(x) say under your prohibitive system of never using antiderivatives. I mean that's what you're doing in integration.

Edit: for all the follow up posts:

if you know something differentiates to give the integrand, you know it is the integral (up to a constant)

that's what integration is.

how did you do the integral of cos(x) first time? or 1 for that matter.

Last edited: Apr 17, 2004
14. Apr 17, 2004

algebrist

but my problem is that i can't integrate
1/sin^2

well, it must be possible to derive it by just knowing the derivatives of sin and cos only.
i tried integration by parts but it would'n work

15. Apr 17, 2004

matt grime

yes, you can integrate it since 1/sin^2 is cosec^2. unless I've imagined the previous 14 posts in my fevered mind. Perhaps you'd just better accept that you ought to have learnt all the trig derivatives? why must it be possible only knowing the derivative of sin and cos? unless the integrand is nice then you cannot do an integral. the vast majority (almost all) integrals do not fall under that heading.

16. Apr 17, 2004

algebrist

anyway......... i see your point i just wanted to do it in a more straight-forward fashion because this antiderivative business seems artificial to me........

with your logic you can just learn by heart the derivatives of infinitely many functions and never use any technique of integration except for antiderivatives......

17. Apr 17, 2004

arildno

Basically, any "technique of integration" is a trick to let us find antiderivatives,
unless you want to abolish the fundamental theorem of calculus and resort to compute the limits of infinite sums in another way.

18. Apr 17, 2004

algebrist

well, i'm always rediscovering the wheel you know...

19. Apr 17, 2004

matt grime

i am not asking you to learn all the derivatives of an infinite number of functions, just the common or garden ones, the elementary ones, and knowing cot's derivative should be one of them, (it's just lilke tan after all). otherwise you couldn't do any integrals. i think cot is a basic one to know, you don't, it appears.

even when you use these techniques, you're only reducing it to a case where you know the anti-derivative, so you ought to at least know some of them.

exercise: find the integral of x^n without usung the notion of an anti derivative.

Last edited: Apr 17, 2004
20. Apr 17, 2004

jdavel

algebrist,

There's not always a turn-key method for finding the integral of a function, the way there is for finding a derivative. Sometimes you have to just feel your way along until something works. Experience and practice help a lot.

If I hadn't known the answer to this problem I think I could have gotten it once it became just the integral of 1/sin(x)^2. That sin(x)^2 in the denominator looks like the last thing you do when you differentiate a ratio (bottom times derivative of the top, minus top times derivative of the bottom, all divided by the bottom squared). So, the integral must be a ratio with sin(x) as the denominator. Then, when I differentiate this integral I'm going to get two terms in the top (bottom times derivative of the top minus....) and their difference has to be 1. Whenever you're working with sines and cosines and you see a 1, think sin^2 + cos^2. And there are my two terms! They add to 1 and since the sin and cos are eachother's derivative and integral, things are looking good. Then I just have to fiddle around with the signs.