Anyone know the integration of 3/(1+2 cos x) dx

Question

how to do the integration of 3/(1+2 cos x) dx

thx!!!!!!

Answer 1

Given ʃ3/[1+2cosx]

let t=tanx/2

dx=2dt/[1+t^2]

ʃ3/[1+2cosx] = 6ʃ1/[3-t^2]dt

Again....... let t=(3)^1/2tanhu

u=tanh^-1(t/3^1/2)

dt=3^1/2sech^2u

6ʃ1/[3-t^2]dt = 6(3)^1/2ʃsech^2u/[3-3tanh^2u]du

=2(3)^1/2ʃudu

=2(3)^1/2u + c

but u=tanh^-1(t/3^1/2)

       =tanh^-1(tan{x/2}/3^1/2)

therefore

Given ʃ3/[1+2cosx]=tanh^-1(tan{x/2}/3^1/2) +c