Question

Integral of cosecx dx =log|tanx/2| show that.​

Answer 1

ANSWER:

Let t = logtan[x/2]

⇒dt = 1/ tan[x/2] * sec² x/2 × ½ dx

⇒dt = 1/2 cos² x/2 × cot x/2dx

⇒dt = 1/2 * 1/ cos² x/2 × cosx/2 / sin x/2 dx

⇒dt = 1/2 cosx/2 / sin x/2 dx

⇒dt = 1/sinxdx

⇒dt = cosecxdx

Putting it in the integration we get,

∫cosecx / log tan(x/2)dx

= ∫dt/t

= log∣t∣+c

= log∣logtan x/2∣+c where t = logtan x/2