`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...

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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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