$$$\cos{\left(\frac{x^{2}}{18} \right)}$$$ 的积分

求$$$\int \cos{\left(\frac{x^{2}}{18} \right)}\, dx$$$。

设$$$u=\frac{\sqrt{2} x}{6}$$$。

则$$$du=\left(\frac{\sqrt{2} x}{6}\right)^{\prime }dx = \frac{\sqrt{2}}{6} dx$$$ (步骤见»)，并有$$$dx = 3 \sqrt{2} du$$$。

因此，

$${\color{red}{\int{\cos{\left(\frac{x^{2}}{18} \right)} d x}}} = {\color{red}{\int{3 \sqrt{2} \cos{\left(u^{2} \right)} d u}}}$$

对 $$$c=3 \sqrt{2}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{3 \sqrt{2} \cos{\left(u^{2} \right)} d u}}} = {\color{red}{\left(3 \sqrt{2} \int{\cos{\left(u^{2} \right)} d u}\right)}}$$

该积分（菲涅耳余弦积分）没有闭式表达式：

$$3 \sqrt{2} {\color{red}{\int{\cos{\left(u^{2} \right)} d u}}} = 3 \sqrt{2} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} u}{\sqrt{\pi}}\right)}{2}\right)}}$$

回忆一下 $$$u=\frac{\sqrt{2} x}{6}$$$:

$$3 \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{u}}}{\sqrt{\pi}}\right) = 3 \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{\left(\frac{\sqrt{2} x}{6}\right)}}}{\sqrt{\pi}}\right)$$

因此，

$$\int{\cos{\left(\frac{x^{2}}{18} \right)} d x} = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right)$$

加上积分常数：

$$\int{\cos{\left(\frac{x^{2}}{18} \right)} d x} = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right)+C$$

$$$\int \cos{\left(\frac{x^{2}}{18} \right)}\, dx = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right) + C$$$A