Find $$\int_0^{t} xe^{i(ax^2)}J_0(bx)\,\mathrm dx$$

Given $a>0,\ b>0, t>0$.

Previously I asked a more difficult integration problem. It seems that it is too difficult to find a solution. Now I omit the $x^4$ term. The problem can be transformed into:

$$\frac{1}{2}\int_0^{t^2} e^{i(a\eta)}J_0(b\sqrt{\eta})\,\mathrm d\eta$$

Where $\eta = x^2$

By checking the integral reference book, I found the following:

$$\int_0^{\infty}J_{0}(\beta x)\sin(\alpha x^2)x\,\mathrm dx=\frac{1}{2\alpha}\cos\left(\frac{\beta^2}{4\alpha}\right),$$

$$\int_0^{\infty}J_{0}(\beta x)\cos(\alpha x^2)x\,\mathrm dx=\frac{1}{2\alpha}\sin\left(\frac{\beta^2}{4\alpha}\right).$$

But the integration range is all the way to $\infty$. I'd like to know if there a solution for a finite integration range $[0, t]$. Thank you!


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