## dinamix one year ago i make some challenge , who answer my question less then 5 min without use google , find this integral or original function of this f(x) = 1/(sin(x)

1. Empty

Oh find the integral of $$f(x)=\csc(x)$$? This is a fun problem it has a neat trick. :D

2. dinamix

$\int\limits_{}^{}\frac{ dx}{ \sin (x) }$

3. ganeshie8

$\int \dfrac{1}{\sin x}\, dx = \int \dfrac{\sin x}{1-\cos^2 x}\, dx = -\int \dfrac{1}{1-u^2}\, du =\cdots$

4. Empty

I guess this trick isn't as systematic as ganehie's but this is how I first learned to do this one: $\int \csc x dx = \int \csc x \frac{\csc x + \cot x}{\csc x + \cot x} dx = \int \frac{\csc ^2 x + \csc x\cot x}{\csc x + \cot x} dx \cdots$

5. dinamix

@ganeshie8 i know the answer and i have easy method

6. ganeshie8

I know that you know :)

7. dinamix

lol nice dude

8. Empty

Ok can we solve this with our arms tied behind our backs? You have to find a new way to solve this.

9. ganeshie8

do you have any other ways... actually the previous trick works for $$\csc^3x$$ too but you will need to love partial fractions to use it : $\int \dfrac{1}{\sin^3 x}\, dx = \int \dfrac{\sin x}{(1-\cos^2 x)^2}\, dx = -\int \dfrac{1}{(1-u^2)^2}\, du =\cdots$

10. dinamix

11. ganeshie8

I like this one : $\int \dfrac{1}{\sin x}\, dx = \int\dfrac{1+\tan^2(x/2)}{2\tan(x/2)}\,dx = \int \dfrac{1}{u}\,du = \log\left(\tan(x/2)\right)+C$

12. dinamix

yeah this is !! u are amazing mate

13. ganeshie8

Ahh thought you have some other clever way..

14. dinamix

yup i have other way

15. ganeshie8

16. dinamix

its good challenge or no ?

17. ganeshie8

sure it is! idk about others, but it really challenged me because i keep forgetting the antiderivatives of cscx and secx and rely on wolfram too much

18. Empty

Yeah same, I had to check real quick: $\frac{d}{dx} (\sin x)^{-1} = - \sin^{-2}x \cos x = -\csc x \cot x$

19. Empty

Also I was trying to see if I could solve it using the infinite product form: $\large \int \csc x dx = \int \frac{1}{x}\prod_{n=1}^\infty \frac{1}{1-\left(\frac{x}{n \pi} \right)} dx$ But nothing really stands out to me.

20. dinamix

but -cscxcotx= log(tan(x/2) + c or no

21. dinamix

http://prntscr.com/88rahv , @ganeshie8 ,@Empty i hope u to understand my solution

22. dinamix

@Empty

23. dinamix

@IrishBoy123

24. IrishBoy123

just watching but now that you ask!!! i looked at this integral recently and found this article: https://en.wikipedia.org/wiki/Integral_of_the_secant_function i am fascinated by the history of maths. often such history is about how something gets discovered and used before it is really understood, eg calculus, complex numbers in this case, the secant integral was just a big thing [secant but same difference] .... and the solution would at the time have been pure rocket science.....

25. IrishBoy123

just my 2 cents :p

26. dinamix

@IrishBoy123 did u see my solution

27. dinamix

@dan815

28. IrishBoy123

yes, it's the same as @ganeshie8 posted above sweet as a nut!

29. dinamix

yup i know this is

30. ikram002p

@IrishBoy123 nuts is something salty xD

31. IrishBoy123

lol!!!! OK, sweet as a nut that has been dipped in honey for a week!

32. dinamix

@IrishBoy123 xD

33. ikram002p

still my joke is better xD