$
J = \int\limits_0^{\frac{\pi }{2}} {\frac{{c{\rm{o}}{{\rm{s}}^3}x}}{{\cos x + 1}}} dx =
\int\limits_0^{\frac{\pi }{2}} {\left( {c{\rm{o}}{{\rm{s}}^2}x - \cos x + 1 - \frac{1}{{\cos x +
1}}} \right)dx} \\
= \int\limits_0^{\frac{\pi }{2}} {\left( {\frac{3}{2} + \frac{1}{2}c{\rm{os}}2x - \cos x -
\frac{1}{{2{{\cos }^2}\frac{x}{2}}}} \right)} dx$
$=(\frac{3}{2}x+\frac{1}{4}\sin 2x-\sin x-\tan\frac{x}{2})|^{\frac{\pi}{2}}_0$
$ = \frac{{3\pi }}{4} - 2
$