dinamix
  • dinamix
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)
Mathematics
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dinamix
  • dinamix
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)
Mathematics
jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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Empty
  • Empty
Oh find the integral of \(f(x)=\csc(x)\)? This is a fun problem it has a neat trick. :D
dinamix
  • dinamix
\[\int\limits_{}^{}\frac{ dx}{ \sin (x) } \]
ganeshie8
  • ganeshie8
\[\int \dfrac{1}{\sin x}\, dx = \int \dfrac{\sin x}{1-\cos^2 x}\, dx = -\int \dfrac{1}{1-u^2}\, du =\cdots \]

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Empty
  • Empty
I guess this trick isn't as systematic as ganehie's but this is how I first learned to do this one: \[\int \csc x dx = \int \csc x \frac{\csc x + \cot x}{\csc x + \cot x} dx = \int \frac{\csc ^2 x + \csc x\cot x}{\csc x + \cot x} dx \cdots \]
dinamix
  • dinamix
@ganeshie8 i know the answer and i have easy method
ganeshie8
  • ganeshie8
I know that you know :)
dinamix
  • dinamix
lol nice dude
Empty
  • Empty
Ok can we solve this with our arms tied behind our backs? You have to find a new way to solve this.
ganeshie8
  • 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 : \[\int \dfrac{1}{\sin^3 x}\, dx = \int \dfrac{\sin x}{(1-\cos^2 x)^2}\, dx = -\int \dfrac{1}{(1-u^2)^2}\, du =\cdots \]
dinamix
  • dinamix
@ganeshie8 so what your solution
ganeshie8
  • ganeshie8
I like this one : \[\int \dfrac{1}{\sin x}\, dx = \int\dfrac{1+\tan^2(x/2)}{2\tan(x/2)}\,dx = \int \dfrac{1}{u}\,du = \log\left(\tan(x/2)\right)+C \]
dinamix
  • dinamix
yeah this is !! u are amazing mate
ganeshie8
  • ganeshie8
Ahh thought you have some other clever way..
dinamix
  • dinamix
yup i have other way
ganeshie8
  • ganeshie8
please do share xD
dinamix
  • dinamix
its good challenge or no ?
ganeshie8
  • 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
Empty
  • Empty
Yeah same, I had to check real quick: \[\frac{d}{dx} (\sin x)^{-1} = - \sin^{-2}x \cos x = -\csc x \cot x\]
Empty
  • 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.
dinamix
  • dinamix
but -cscxcotx= log(tan(x/2) + c or no
dinamix
  • dinamix
http://prntscr.com/88rahv , @ganeshie8 ,@Empty i hope u to understand my solution
dinamix
  • dinamix
dinamix
  • dinamix
IrishBoy123
  • 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.....
IrishBoy123
  • IrishBoy123
just my 2 cents :p
dinamix
  • dinamix
@IrishBoy123 did u see my solution
dinamix
  • dinamix
IrishBoy123
  • IrishBoy123
yes, it's the same as @ganeshie8 posted above sweet as a nut!
dinamix
  • dinamix
yup i know this is
ikram002p
  • ikram002p
@IrishBoy123 nuts is something salty xD
IrishBoy123
  • IrishBoy123
lol!!!! OK, sweet as a nut that has been dipped in honey for a week!
dinamix
  • dinamix
ikram002p
  • ikram002p
still my joke is better xD

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