Integrating cot(x) to the fourth
By Herbert Pang (p242) on October 29, 1998 :

I cannot solve one of the integration question in my textbook. Can you please help?

òcot4 x dx
Thanks for your attention.
By Richard (rpd25) on October 30, 1998 :

Dear Herbert,



If this is the only point you get stuck at in the evaluation of integrals then you obviously have a very good grasp of the methods involved, and this is merely an annoying example. I bashed away at this for 10 minutes before my good friend Rosemary Bell (to whom I take all my really tricky problems), pointed out the following solution:

(cot x)4 = (1-(sin x)2 )2 /(sin x)4
= (cosec x)4 - 2(cosec x)2 + 1 (expanding)


Now, (cosec x)4 = (1 + (cot x)2 )(cosec x)2
= (cosec x)2 + (cosec x)2 (cot x)2

So,
(cot x)4 = (cosec x)2 *cot x)2 - (cosec x)2 + 1

Bearing in mind that d(cot x)/dx = -(cosec x)2 , we can now see by inspection that the integral of (cot x)4 is:
-1/3 (cot x)3 + (cot x) + x + c
where c is an arbitrary constant (I know how pedantic A-level maths teachers are about that).

And there you have it. I'm not sure it's terribly instructive but it is a solution. Now go and do some interesting maths. :-)

Best wishes,
Richard.