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`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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