help with an Integral

[akleron]

Hello !
I'm trying to calculate the following integral but kinda stuck.
there is a hint in the answers (in the 2nd picture) which i tried to use.
Any advice ? did I started right ? what should I do from here ?
thank you !


the question:


the answer:


my latest attempt:

 

[Cubist]

A slight mistake... you missed something here:-


I managed to show that:-
[math] \frac{2}{1+t^2+a \left(1-t^2\right)} = \frac{1+\cos\left(x\right)}{1+a\cdot \cos\left(x\right)} [/math]
But I can't immediately see how to integrate the LHS of the above equation

EDIT: I should probably give you a hint about how to show the equivalence above. Rewrite tan^2(x/2) as (1-cos(x))/(1+cos(x))

 

[akleron]

A slight mistake... you missed something here:-
View attachment 19062

I managed to show that:-
[math] \frac{2}{1+t^2+a \left(1-t^2\right)} = \frac{1+\cos\left(x\right)}{1+a\cdot \cos\left(x\right)} [/math]
But I can't immediately see how to integrate the LHS of the above equation

EDIT: I should probably give you a hint about how to show the equivalence above. Rewrite tan^2(x/2) as (1-cos(x))/(1+cos(x))

Ohh I forgot to multiply it by 1/2 ...
I will try again with your hint
Thank you