kill(all)$

f(x):=factor(trigsimp(sqrt(cos(2*x)^2*(5-sin(2*x)))))$ f(x);

F(x):=(-1/3)*(5-sin(2*x))^(3/2)*signum(cos(2*x));

f_(x):=sqrt(5)*abs(cos(2*x));

F_(x):=sqrt(5)*sin(x)*cos(x)*signum(cos(2*x));

F(%pi/2)-F(0); float(%);

F_(%pi/2)-F_(0); float(%);

     Numerical Solution

quad_qags(f(x),x,0,%pi/2);

quad_qags(f_(x),x,0,%pi/2);