I'm attempting to find the leading behavior of the integral
$$I(x) = \int_{x}^{1} \cos(xt)dt ~~~~~~~~~~~~~~~~~ x \rightarrow 0^+.$$
My attempt:
$\cos(xt)$ has Taylor Series expansion centered at $x = 0$ given by
$$\sum_{n=0}^{\infty}\frac{(-1)^n(tx)^{2n}}{(2n)!}$$
and if this series is uniformly asymptotic to $\cos(xt)$ for $t \in [x,1]$ as $x \rightarrow 0^+$, then the integral is asymptotic to the result of term by term integration of the series, i.e.,
$\begin{align} \int_x^1 \cos(xt)dt ~~~ &\text{(tilde)} ~~~\sum_{n=0}^{\infty}x^{2n} \int_x^1 \frac{(-1)^nt^{2n}}{(2n)!} \\ & \text{(tilde)} ~~~ \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}(1-x^{2n+1})}{(2n+1)!} \end{align}$
My question is, how do I conclude uniform convergence of the series to apply this reasoning? Will it converge for only certain values of $x$, as well as $t$?