The value of the integral $\int \frac{x \sin x^2 e^{\sec x^2}}{\cos ^2 x^2} d x$, is

$\frac{1}{2} e^{\sec x^2}+C$

Let

$I =\int \frac{x \sin x^2 e^{\sec x^2}}{\cos ^2 x^2} d x=\frac{1}{2} \int e^{\sec x^2} . 2 x \tan x^2 \sec x^2 d x$

$\Rightarrow I =\frac{1}{2} \int e^{\sec x^2} d\left(\sec x^2\right)=\frac{1}{2} e^{\sec x^2}+C$