Evaluate integral 2/(1-cos2x )dx

integral 2/(1-cos2x)dx = 2integral 1/(1-cos2x)dx

or integral 2/(1-cos2x)dx = 2integral 1/[2sin²(x/2)]dx

or integral 2/(1-cos2x)dx = 2/2integral 1/[sin²(x/2)]dx

or integral 2/(1-cos2x)dx = integral cosec²(x/2)dx

Let x/2 = t

so, dx = 2dt.

So, integral 2/(1-cos2x)dx = integral cosec²(x/2)dx = integral cosec²t(2dt)

or integral 2/(1-cos2x)dx = integral 2cosec²tdt

or integral 2/(1-cos2x)dx = 2integral cosec²tdt

or integral 2/(1-cos2x)dx = 2(-cot t)

or integral 2/(1-cos2x)dx = -2cot(x/2).

Hence, integral 2/(1-cos2x)dx = -2cot(x/2).