$\int cosec (x) dx$

By Anonymous on Friday, February 9, 2001 - 09:57 pm :

How can you integrate the following:
$\int cosec (x) dx$ ??
Thanks

By James Lingard (Jchl2) on Friday, February 9, 2001 - 10:35 pm :

Multiply the integrand by

(cosec (x) + \cot (x)) / (cosec (x) + \cot (x))

. Then the numerator is the derivative of the denominator, so you get:

$\int cosec (x) dx = - \ln (cosec (x) + \cot (x)) + c$

James.

(Method supplied by two of my physicist and chemical engineer friends who are considerably better than me at this sort of thing!)

By Anonymous on Friday, February 9, 2001 - 10:58 pm :

Very neat James. Thank you.
Its like when you have to integrate:
$\int \sec (x) dx$ , and you have to multiply top and bottom by $(\sec (x) + \tan (x))$ .
Why does this method work, I mean, unless you've been shown it before, how would you have come to the conclusion of multiplying top and bottom by it?

Thanks.

PS: also thanks to your friends! ;)

By James Lingard (Jchl2) on Friday, February 9, 2001 - 11:06 pm :

Don't ask me - I don't think I'd ever have come up with it!

James.