Problem: Consider the following integral:$I(x)=\int_x^{1}cos(xt)dt$In the limit$x \to 0+$, find approximate behavior of the integral up to and including the order x^6. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x) = 1 - x - \frac{x^2}{6} + \frac{x^4}{120} + \frac{x^5}{6} - \frac{x^6}{5040} }$