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$$$\sin{\left(x^{2} - 4 \right)}$$$の積分
入力内容
$$$\int \sin{\left(x^{2} - 4 \right)}\, dx$$$ を求めよ。
解答
被積分関数を書き換える:
$${\color{red}{\int{\sin{\left(x^{2} - 4 \right)} d x}}} = {\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(4 \right)} - \sin{\left(4 \right)} \cos{\left(x^{2} \right)}\right)d x}}}$$
項別に積分せよ:
$${\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(4 \right)} - \sin{\left(4 \right)} \cos{\left(x^{2} \right)}\right)d x}}} = {\color{red}{\left(- \int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x} + \int{\sin{\left(x^{2} \right)} \cos{\left(4 \right)} d x}\right)}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\cos{\left(4 \right)}$$$ と $$$f{\left(x \right)} = \sin{\left(x^{2} \right)}$$$ に対して適用する:
$$- \int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\int{\sin{\left(x^{2} \right)} \cos{\left(4 \right)} d x}}} = - \int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\cos{\left(4 \right)} \int{\sin{\left(x^{2} \right)} d x}}}$$
この積分（フレネル正弦積分）には閉形式はありません:
$$- \int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x} + \cos{\left(4 \right)} {\color{red}{\int{\sin{\left(x^{2} \right)} d x}}} = - \int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x} + \cos{\left(4 \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\sin{\left(4 \right)}$$$ と $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$ に対して適用する:
$$\frac{\sqrt{2} \sqrt{\pi} \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} - {\color{red}{\int{\sin{\left(4 \right)} \cos{\left(x^{2} \right)} d x}}} = \frac{\sqrt{2} \sqrt{\pi} \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} - {\color{red}{\sin{\left(4 \right)} \int{\cos{\left(x^{2} \right)} d x}}}$$
この積分（フレネル余弦積分）には閉形式はありません:
$$\frac{\sqrt{2} \sqrt{\pi} \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} - \sin{\left(4 \right)} {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} = \frac{\sqrt{2} \sqrt{\pi} \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} - \sin{\left(4 \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$
したがって、
$$\int{\sin{\left(x^{2} - 4 \right)} d x} = - \frac{\sqrt{2} \sqrt{\pi} \sin{\left(4 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$$
簡単化せよ:
$$\int{\sin{\left(x^{2} - 4 \right)} d x} = \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(4 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}$$
積分定数を加える:
$$\int{\sin{\left(x^{2} - 4 \right)} d x} = \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(4 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}+C$$
解答
$$$\int \sin{\left(x^{2} - 4 \right)}\, dx = \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(4 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(4 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2} + C$$$A