Consider the Fresnel integral
$$C(x) = \int_0^x \cos(0.5\pi t^2) \ dt.$$
I've calculated that, as $x\to 0$, $C(x) \sim x$ and, as $x\to\infty$, $C(x) \sim 0.5 + \frac{\sin(0.5\pi x^2)}{\pi x}$. Using these asymptotic expansions, is it possible to then deduce an approximation to $C(x)$ that is valid for all $x > 0$?
$$C(x) = \int_0^x \cos \left(\frac{\pi t^2}{2}\right) \, dt$$ Use the fundamental theorem of calculus $$C'(x)=\cos \left(\frac{\pi x^2}{2}\right)$$ and the expansion of $\cos(y)$ is valid for all $y$. Use it, replace and integrate termwise for all $x$. $$C(x)=\sum_{n=0}^\infty (-1)^n \frac{ \left(\frac\pi{2}\right)^{2 n}} {(4n+1)\, (2n)! }x^{2n+1}$$