关于将积分∫₀ˣsin(α+t²)dt转化为单一形式的咨询

Hey there! I get that you're looking to rewrite the integral $\int_0^{x\sin{(\alpha+t}2)}dt$ as a single term instead of the linear combination of Fresnel integrals $C(x)$ and $S(x)$ you already derived. Let's break down the options you have here:

1. Using the Complex Fresnel Integral #

If you're open to working with complex-valued special functions, this is the most straightforward way to get a single-term expression. First, recall Euler's formula for sine:
$$
\sin(\alpha + t^2) = \frac{e^{i(\alpha + t^2)} - e^{-i(\alpha + t^2)}}{2i}
$$

Substitute this into your integral:
$$
\int_0^x \sin(\alpha + t^2)dt = \frac{e^{i\alpha}}{2i} \int_0^x e^{it2}dt - \frac{e^{-i\alpha}}{2i} \int_0^x e^{-it2}dt
$$

The complex Fresnel integral is typically defined as:
$$
F(x) = \int_0^x e^{it2}dt = C(x) + iS(x)
$$
Its complex conjugate $\overline{F(x)} = C(x) - iS(x)$ equals $\int_0^x e^{-it2}dt$. Plugging these back in, your integral simplifies to:
$$
\frac{e^{i\alpha}F(x) - e^{-i\alpha}\overline{F(x)}}{2i}
$$
This is a single-term expression, leveraging the complex Fresnel integral.

2. Defining a Custom Special Function #

If you want to stick to real-valued functions but still have a single term, you can simply define a new function tailored to your integral:
$$
S_\alpha(x) = \int_0^x \sin(\alpha + t^2)dt
$$
This wraps the combination of $C(x)$ and $S(x)$ into a single symbol, which is useful if you're going to work with this integral repeatedly. The tradeoff is that this isn't a standard special function, so others might need your definition to understand it.

3. Why a Single Real Standard Special Function Isn't Feasible #

It’s worth noting that there’s no way to express this integral using only one of the standard real Fresnel integrals $C(x)$ or $S(x)$. The original integrand $\sin(\alpha + t^2)$ expands to a linear combination of $\cos(t^2)$ and $\sin(t^2)$, whose antiderivatives are exactly $C(x)$ and $S(x)$. Any real-valued expression will inherently be a combination of these two—there’s no standard real special function that directly captures this specific linear combination for arbitrary $\alpha$.

Hope these options give you a clear path forward!

备注：内容来源于stack exchange，提问作者Xiangyu Cui