Answer
$\frac{\sin \left(1\right)}{3}$
Work Step by Step
$\int _0^1\int _0^{s^2}\cos \left(x^3\right)dtds$
$\int _0^{x^2}\cos \left(s^3\right)dt=s^2\cos \left(s^3\right)$
Note: $\int \:x^2\cos \left(x^3\right)dx=\frac{1}{3}\sin \left(x^3\right)+C$
So,
$\int _0^1s^2\cos \left(s^3\right)dx=\frac{\sin \left(1\right)}{3}$
