Evaluate the following integrals:

∫ cosec x log (cosec x – cot x) dx

Assume log(cosec x – cot x) = t

d(log(cosec x – cot x)) = dt

(use chain rule to differentiate first differentiate log(secx + tanx) then (secx + tanx)

⇒ = dt

⇒

⇒ cscx dx = dt

Put t and dt in given equation we get

Substituting the values oft and dt we get

⇒

⇒

But t = log(cosec x – cot x)

⇒ .