Evaluating definite integrals using the fundamental theorem of calculus.

Problem

Consider the functions f and F, where f, left parenthesis, x, right parenthesis, equals, start fraction, 1, plus, cosine, start superscript, 2, end superscript, x, divided by, cosine, start superscript, 2, end superscript, x, end fraction, space and F, prime, left parenthesis, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, space.
If space, F, left parenthesis, start fraction, pi, divided by, 4, end fraction, right parenthesis, equals, start fraction, pi, divided by, 2, end fraction, space, then what is space, F, left parenthesis, 0, right parenthesis, space?
Please choose from one of the following options.