int_(-pi/2)^(pi/2) cosx/(1+sin^2x) dx Use integration tables to evaluate the definite integral.

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

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