# Computing Asymptotics of a Fresnel Integral At Infinity

I am very new to Mathematica, and am trying to write code that will give me the series asymptotics for the function $$F(y) = \int_0^y f(y)\;dy,$$ where $$f(y) = y^2 \cos(y^2) C( (2/\pi)^{1/2} y)^2$$, as $$y \to \infty$$. This is mainly an exercise so I can learn how to compute things in Mathematica. To compute the asymptotics, I note that if $$k(x) = f(1/x)$$, then $$F(1/y) = \int_0^{1/y} f(y)\; dy = \int_0^x k(x) (-1/x^2)\; dx.$$ Thus I just have to compute the asymptotics of the function $$K(x) = F(1/x)$$ as $$x \to 0$$, where $$K$$ is the right hand side above. I am able to get some expression for the asymptotics using the code below, but I would prefer the asymptotics to not include implicit antiderivatives. Am I doing something stupid?

Here's my code:

f[y_] := y^2*Cos[y^2]*FresnelC[Sqrt[2/Pi]*y]^2;
antif[y_] := Integrate[f[y], y];
k[x_] := a  /. Solve[{a == f[y], x*y == 1}, {a, y}];
antik[x_] := Integrate[k[x]*(-1/x^2), x]
Series[antik[x], {x, 0, 5}] // TraditionalForm


And here is an image of the asymptotics I obtain New contributor
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