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Put cos 2x = 1 / t
So, dx = dt / ( 2t2 . sin 2x)
so,  cos2x/sinx = dt / ( 2t2 . 2sin2x) * t * cosx = dt / 2 * t(t-1) t+1
Now put t + 1 = p2
So,  cos2x/sinx = 2* dp / (p2 - 1)*( p2 - 2 )
= 2* dp / ( p2 - 2 ) - 2* dp / ( p2 - 1 )
Now these integrals can be easily intergated and thus we are done by resubstituting the values of p.
Cheers !!!!!!!!!!!!
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