Split the integral at a constant x = c and expand it as
[tex]\displaystyle \int_x^{x^2} \sin(t) \, dt = \int_x^c \sin(t) \, dt + \int_c^{x^2} \sin(t) \, dt \\\\ = \int_c^{x^2} \sin(t) \, dt - \int_c^x \sin(t) \, dt[/tex]
Now apply the fundamental theorem of calculus.
[tex]\displaystyle \frac{d}{dx} \int_c^{x^2} \sin(t) \, dt = \sin(x^2) \frac{d}{dx} x^2 = 2x \sin(x^2)[/tex]
[tex]\displaystyle \frac{d}{dx} \int_c^x \sin(t) \, dt = \sin(x)[/tex]
and so
[tex]\displaystyle \frac{d}{dx} \int_x^{x^2} \sin(t) \, dt = \boxed{2x \sin(x^2) - \sin(x)}[/tex]
