Question:
integral from pi to 0 of

dx/(2+cos x)

i used the substitution t=tan(x/2) and i ended up with

integral from +infinity to 0 of

2dt/(t²+3)

which looks like an inverse tan function , and i ended up with sqr(27)/2 pi , which is not the same as my calculator's answer , so i suspct i am doing some thing wrong. can some one tell me where i am going wrong please.

I would do this integral exactly as you did with a t=tan(x/2) substitution and then an inverse tangent. This gives me

Evaluating this from pi to 0 I get  ^-pi/_sqrt(3)

You can check the integration using Mathematica at The Integrator. You need to take some care with Mathematica's notation. Input the expression