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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)
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)

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Empty
 one year ago
Best ResponseYou've already chosen the best response.2Oh find the integral of \(f(x)=\csc(x)\)? This is a fun problem it has a neat trick. :D

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3\[\int\limits_{}^{}\frac{ dx}{ \sin (x) } \]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4\[\int \dfrac{1}{\sin x}\, dx = \int \dfrac{\sin x}{1\cos^2 x}\, dx = \int \dfrac{1}{1u^2}\, du =\cdots \]

Empty
 one year ago
Best ResponseYou've already chosen the best response.2I guess this trick isn't as systematic as ganehie's but this is how I first learned to do this one: \[\int \csc x dx = \int \csc x \frac{\csc x + \cot x}{\csc x + \cot x} dx = \int \frac{\csc ^2 x + \csc x\cot x}{\csc x + \cot x} dx \cdots \]

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3@ganeshie8 i know the answer and i have easy method

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4I know that you know :)

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Ok can we solve this with our arms tied behind our backs? You have to find a new way to solve this.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4do 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 : \[\int \dfrac{1}{\sin^3 x}\, dx = \int \dfrac{\sin x}{(1\cos^2 x)^2}\, dx = \int \dfrac{1}{(1u^2)^2}\, du =\cdots \]

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3@ganeshie8 so what your solution

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4I like this one : \[\int \dfrac{1}{\sin x}\, dx = \int\dfrac{1+\tan^2(x/2)}{2\tan(x/2)}\,dx = \int \dfrac{1}{u}\,du = \log\left(\tan(x/2)\right)+C \]

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3yeah this is !! u are amazing mate

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4Ahh thought you have some other clever way..

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3its good challenge or no ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4sure 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

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Yeah same, I had to check real quick: \[\frac{d}{dx} (\sin x)^{1} =  \sin^{2}x \cos x = \csc x \cot x\]

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Also 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.

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3but cscxcotx= log(tan(x/2) + c or no

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3http://prntscr.com/88rahv , @ganeshie8 ,@Empty i hope u to understand my solution

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1just 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.....

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1just my 2 cents :p

dinamix
 one year ago
Best ResponseYou've already chosen the best response.3@IrishBoy123 did u see my solution

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1yes, it's the same as @ganeshie8 posted above sweet as a nut!

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.0@IrishBoy123 nuts is something salty xD

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1lol!!!! OK, sweet as a nut that has been dipped in honey for a week!

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.0still my joke is better xD
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