Answer
$$\frac{1}{2}\tan \left( {{x^2}} \right) + C$$
Work Step by Step
$$\eqalign{
& {\text{substitute }}u = {x^2},{\text{ }}du = 2xdx \cr
& \int {{{\sec }^2}} \left( {{x^2}} \right)xdx \cr
& = \int {{{\sec }^2}} u\left( {\frac{1}{2}du} \right) \cr
& = \frac{1}{2}\int {{{\sec }^2}u} du \cr
& {\text{find the antiderivative }} \cr
& = \frac{1}{2}\tan u + C \cr
& {\text{write in terms of }}x \cr
& = \frac{1}{2}\tan \left( {{x^2}} \right) + C \cr} $$
