O primitiva (1)

[stuart clark]

$\displaystyle \int\frac{(1+\cos 2x)}{(2+\cos x)^2}dx$

[stuart clark]

$\displaystyle \int \frac {(1 + \cos 2x)} {(2 + \cos x) ^ 2} dx$

$\displaystyle \int\frac{2\cos^2 (x)}{\left(2+\cos x\right)^2}dx$

Put $\displaystyle \cos x = \frac{1-\tan^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}$

$\displaystyle 2\int\frac{\left(1-\tan^2 \frac{x}{2}\right)^2}{\left(3+\tan^2 \frac{x}{2} \right)^2}dx$

Let $\tan \frac{x}{2} = t\Leftrightarrow \sec^2 \frac{x}{2} = 2dt$

$\displaystyle 4\int\frac{(1-t^2)^2}{(3+t^2)^2.(1+t^2)}dt$

$\displaystyle 4\int\frac{1}{1+t^2}dt - 32\int\frac{1}{(3+t^2)^2}dt$

$4\tan^{-1}(t) - \frac{16\sqrt{3}}{9}.\tan^{-1}\left(\frac{t}{\sqrt{3}}\right)+\frac{96}{9}.\frac{1}{t^2+9}+C$

$4\tan^{-1}{\tan \left(\frac{x}{2}\right) - \frac{16\sqrt{3}}{9}.\tan^{-1}\left(\frac{\tan \frac{x}{2}}{\sqrt{3}}\right)+\frac{96}{9}.\frac{1}{\tan ^2\frac{x}{2}+9}+C$