Problem: Write the first two term asymptotic series of $I(x) = \int^x_1 \ln(xt^2)\cos(t^3) dt$ in the limit $x \rightarrow \infty$. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x) = \frac{\ln(x^3)\sin(x^3)}{3x^2} - \frac{\ln(x)\sin(1)}{3} -\frac{2(\ln(x^3)-1)\cos(x^3)}{9x^5} + \frac{2(\ln(x)-1)\cos(1)}{9}}$