Double integral inequality

Question

Proof this inequality:

$$\int_1^4 \int_0^1 (x^2+\sqrt{y})\cos(x^2y^2) dx dy\leq 9 $$

I don't know how to approach to this, any idea ?

Answer 1

Hint: $0 < x < 1$ and $1 < y < 4$ implies $|(x^2 + \sqrt{y}) \cos(x^2 y^2)| \le \ldots$.