`int_0^(pi/3) (sin(theta) + sin(theta)tan^2(theta))/(sec^2(theta)) d theta` Evaluate the integral

Before evaluating this integral, simplify the expression in the integral using trigonometric identities. The following Pythagorean identity will be useful:

`tan^2(theta) + 1 = sec^2(theta)`

Start by factoring out `sin(theta) ` from the numerator of the fraction:

`(sin(theta) + sin(theta)tan^2(theta))/(sec^2(theta)) = (sin(theta)(1 + tan^2(theta)))/(sec^2(theta))`

Since `1 + tan^2(theta) = sec^2(theta)` , the parenthesis in the numerator cancels, so the fraction equals `sin(theta)` .

Then the given integral becomes

`int _0 ^ (pi/3) sin(theta)d(theta) =...

Before evaluating this integral, simplify the expression in the integral using trigonometric identities. The following Pythagorean identity will be useful:

`tan^2(theta) + 1 = sec^2(theta)`

Start by factoring out `sin(theta) ` from the numerator of the fraction:

`(sin(theta) + sin(theta)tan^2(theta))/(sec^2(theta)) = (sin(theta)(1 + tan^2(theta)))/(sec^2(theta))`

Since `1 + tan^2(theta) = sec^2(theta)` , the parenthesis in the numerator cancels, so the fraction equals `sin(theta)` .

Then the given integral becomes

`int _0 ^ (pi/3) sin(theta)d(theta) = -cos(theta) |_0 ^ (pi/3) = -(cos(pi/3) - cos(0)) = -(1/2 - 1) = 1/2`

The given integral equals `1/2` .