Find the intergral of the following:-

1/(1+cotx) =sinx/(sinx+cosx) =1/2  (   1+ (sinx-cosx)(sinx+cosx) )

Now I= ∫1/2  (   1+ (sinx-cosx)(sinx+cosx) )dx=x/2 +1/2  ∫  (sinx-cosx)(sinx+cosx)  dx

Put sinx+cosx=t in the second term,

1/2  ∫  (sinx-cosx)(sinx+cosx)  dx=1/2   ∫  -dt/t = -1/2lnt

Hence I=x/2-1/2ln(sinx+cosx) +c

Similarily second can be solved.