$$$\sin{\left(x^{2} + y \right)}$$$ の $$$x$$$ に関する積分

$$$\int \sin{\left(x^{2} + y \right)}\, dx$$$ を求めよ。

被積分関数を書き換える:

$${\color{red}{\int{\sin{\left(x^{2} + y \right)} d x}}} = {\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x^{2} \right)}\right)d x}}}$$

項別に積分せよ:

$${\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x^{2} \right)}\right)d x}}} = {\color{red}{\left(\int{\sin{\left(x^{2} \right)} \cos{\left(y \right)} d x} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\cos{\left(y \right)}$$$ と $$$f{\left(x \right)} = \sin{\left(x^{2} \right)}$$$ に対して適用する:

$$\int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\int{\sin{\left(x^{2} \right)} \cos{\left(y \right)} d x}}} = \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\cos{\left(y \right)} \int{\sin{\left(x^{2} \right)} d x}}}$$

この積分（フレネル正弦積分）には閉形式はありません:

$$\cos{\left(y \right)} {\color{red}{\int{\sin{\left(x^{2} \right)} d x}}} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} = \cos{\left(y \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\sin{\left(y \right)}$$$ と $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$ に対して適用する:

$$\frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + {\color{red}{\int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}}} = \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + {\color{red}{\sin{\left(y \right)} \int{\cos{\left(x^{2} \right)} d x}}}$$

この積分（フレネル余弦積分）には閉形式はありません:

$$\sin{\left(y \right)} {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} = \sin{\left(y \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$$

したがって、

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

簡単化せよ:

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

積分定数を加える:

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

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