Question

integration (cosec 2x)^n dx​

Answer 1

cosec 2x = -(-cosex 2x cot 2x -cosec² 2x)/(cosec 2x - cot 2x)

= derivative/function => integral = log (function)

=> integral (cosec 2x) = -1/2 log [A(cosec 2x - cot 2x)] = log [A(cosec2x - cot2x)]^-1/2

1/2*(cosec(2x) - cot(2x)),

The another way I tried :

cosecx = 1/(sin2x)

= 1/(2sinxcosx)

Then used sinx = tanxcosx and substitued this into eqn:

= 1/(2cosxtanxcosx)

= 1/(2tanx (cosx)^2))

= 0.5 ((secx)^2/tanx)

To find integral use substitution method as differential of tanx = (secx)^2

so final ans

integral (cosec2x) = 0.5ln(tanx)

I don't know whether that is the right ans - but does that make sense?!?