The curve reaches a minimum at x=0 and a maximum at x=pi/2. Assuming that the mean value is midway between the values at these two points:

Mean = (0 + cos((1))^2 + (sin(1))^2 + (cos(0))^2)/2

     = (0 + 1 + 1)/2

     = 1

Integral = 1 * pi/2 = pi/2.

Do we have a proof that the curve is symmetrical about the horizontal line y = 1, that is, the mean between the maximum and the minimum? Well, the function evaluated at pi/4+.2 is 1.2725591 and at pi/4-.2 is .7274409, averaging 1.  There's no reason to think choices of the delta other than .2 would not also verify the symmetry of the curve.

Final answer: pi/2