Question

∫(1 – cosx)cosec²xdx = _________ + c 

(a) tan(x/2) 

(b) cot(x/2) 

(c) (1/2)tan(x/2) 

(d) 2tan(x/2)

Answer 1

The correct option (a) tan(x/2)   

Explanation:

(1 – cosx)cose²x = cosec²x – cosx ∙ cosec²x 

= cosec²x – cosecx ∙ cotx 

∴ I = ∫(cosec²x – cosecxcotx)dx 

= – cotx + cosecx + c 

= [(– cosx + 1)/(sinx)] + c 

= [(1 – cosx)/(sinx)] + c 

= [{2sin²(x/2)}/{2sin(x/2) ∙ cos(x/2)}] + c 

= tan(x/2) + c