Question

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)

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

I guess this trick isn't as systematic as ganehie's but this is how I first learned to do this one:

\[\int \csc x dx = \int \csc x \frac{\csc x + \cot x}{\csc x + \cot x} dx = \int \frac{\csc ^2 x + \csc x\cot x}{\csc x + \cot x} dx \cdots \]

Answer 5

@ganeshie8 i know the answer and i have easy method

Answer 6

I know that you know :)

Answer 7

lol nice dude

Answer 8

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

Answer 9

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 :

\[\int \dfrac{1}{\sin^3 x}\, dx = \int \dfrac{\sin x}{(1-\cos^2 x)^2}\, dx = -\int \dfrac{1}{(1-u^2)^2}\, du =\cdots \]

Answer 10

@ganeshie8 so what your solution

Answer 11

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

Answer 12

yeah this is !! u are amazing mate

Answer 13

Ahh thought you have some other clever way..

Answer 14

yup i have other way

Answer 15

please do share xD

Answer 16

its good challenge or no ?

Answer 17

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

Answer 18

Yeah same, I had to check real quick:

\[\frac{d}{dx} (\sin x)^{-1} = - \sin^{-2}x \cos x = -\csc x \cot x\]

Answer 19

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.

Answer 20

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

Answer 21

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

Answer 22

@Empty

Answer 23

@IrishBoy123

Answer 24

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.....

Answer 25

just my 2 cents :p

Answer 26

@IrishBoy123 did u see my solution

Answer 27

@dan815

Answer 28

yes, it's the same as @ganeshie8 posted above

sweet as a nut!

Answer 29

yup i know this is

Answer 30

@IrishBoy123 nuts is something salty xD

Answer 31

lol!!!!

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

Answer 32

@IrishBoy123 xD

Answer 33

still my joke is better xD