A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

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)

  • This Question is Closed
  1. Empty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  2. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  3. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

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

  4. Empty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @ganeshie8 i know the answer and i have easy method

  6. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    I know that you know :)

  7. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    lol nice dude

  8. Empty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  9. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @ganeshie8 so what your solution

  11. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    yeah this is !! u are amazing mate

  13. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    Ahh thought you have some other clever way..

  14. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    yup i have other way

  15. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    please do share xD

  16. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    its good challenge or no ?

  17. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  21. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  22. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @Empty

  23. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @IrishBoy123

  24. IrishBoy123
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    just my 2 cents :p

  26. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @IrishBoy123 did u see my solution

  27. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @dan815

  28. IrishBoy123
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  29. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    yup i know this is

  30. ikram002p
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @IrishBoy123 nuts is something salty xD

  31. IrishBoy123
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  32. dinamix
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @IrishBoy123 xD

  33. ikram002p
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    still my joke is better xD

  34. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.