Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 43

For the given integral problem: int_(-pi/2)^(pi/2) cos(x)/(1+sin^2(x)) dx , we can evaluate this applying indefinite integral formula: int f(x) dx = F(x) +C
where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as the constant of integration.
From the basic indefinite integration table, the problem resembles one of the formula for integral of rational function:
int (du)/(1+u^2)= arctan (u) +C .
For easier comparison, we may apply u-substitution by letting: u = sin(x) then du =cos(x) dx . Since x=+-pi/2 then u=+-1
int_(-pi/2)^(pi/2) cos(x)/(1+sin^2(x)) dx
=int_-1^1 (du)/(1+u^2)
= arctan(u) |_-1^1
=arctan(1)-arctan(-1)
=pi/4- (-pi/4)
=pi/4+pi/4
=(2pi)/4
= pi/2