Well I've heard of Leibniz integral rule but don't quite understand it. Other than that I've no idea for how this can be solved given that $\mbox{sinc}(x)$ is an improper integral.
Also, I've used Wolfram Alpha and it states that the solution is $2x\mbox{sinc}(x^2)-\mbox{sinc}(x)$ but I've no idea how it computed the answer?
Hint. You probably know the Fundamental Theorem of Calculus: if $f$ is a continuous function in $[x_0, x]$ then $$\frac{d}{dx}\int_\limits{x_0}^{x}f(t) dt=f(x).$$ Moreover, note that $$\frac{d}{dx}\int_\limits{x}^{x^2}{\frac{\sin t}t\,dt}= \frac{d}{dx}\int_\limits{1}^{x^2}{\frac{\sin t}t\,dt}-\frac{d}{dx}\int_\limits{1}^{x}{\frac{\sin t}t\,dt}.$$ Now be careful with the first derivative.