\( \int _{}^{}\frac{1}{1+\sin (x)^{2}} d x \)

To evaluate the integral \( \int \frac{1}{1+\sin^2(x)} \, dx \), you can utilize a trigonometric identity and a substitution. A common approach is to use the Weierstrass substitution, \( t = \tan\left(\frac{x}{2}\right) \), which transforms sine and cosine functions into a rational form. This substitution leads to simplifying the integral into a more manageable form involving \( t \). Another option for solving this integral is to use the fact that \( 1 + \sin^2(x) = \frac{2 + \cos(2x)}{2} \) and thereby rewrite the integral. By doing so, you can separate it into a simpler integral that can be evaluated using standard techniques. Happy integrating!