$$$\sin{\left(x^{2} + y \right)}$$$ 关于$$$x$$$的积分
您的输入
求$$$\int \sin{\left(x^{2} + y \right)}\, dx$$$。
解答
改写被积函数:
$${\color{red}{\int{\sin{\left(x^{2} + y \right)} d x}}} = {\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x^{2} \right)}\right)d x}}}$$
逐项积分：
$${\color{red}{\int{\left(\sin{\left(x^{2} \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x^{2} \right)}\right)d x}}} = {\color{red}{\left(\int{\sin{\left(x^{2} \right)} \cos{\left(y \right)} d x} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}\right)}}$$
对 $$$c=\cos{\left(y \right)}$$$ 和 $$$f{\left(x \right)} = \sin{\left(x^{2} \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$\int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\int{\sin{\left(x^{2} \right)} \cos{\left(y \right)} d x}}} = \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} + {\color{red}{\cos{\left(y \right)} \int{\sin{\left(x^{2} \right)} d x}}}$$
该积分（菲涅耳正弦积分）没有闭式表达式：
$$\cos{\left(y \right)} {\color{red}{\int{\sin{\left(x^{2} \right)} d x}}} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x} = \cos{\left(y \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}} + \int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}$$
对 $$$c=\sin{\left(y \right)}$$$ 和 $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$\frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + {\color{red}{\int{\sin{\left(y \right)} \cos{\left(x^{2} \right)} d x}}} = \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + {\color{red}{\sin{\left(y \right)} \int{\cos{\left(x^{2} \right)} d x}}}$$
该积分（菲涅耳余弦积分）没有闭式表达式：
$$\sin{\left(y \right)} {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} = \sin{\left(y \right)} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$$
因此，
$$\int{\sin{\left(x^{2} + y \right)} d x} = \frac{\sqrt{2} \sqrt{\pi} \sin{\left(y \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$$
化简：
$$\int{\sin{\left(x^{2} + y \right)} d x} = \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(y \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}$$
加上积分常数：
$$\int{\sin{\left(x^{2} + y \right)} d x} = \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(y \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}+C$$
答案
$$$\int \sin{\left(x^{2} + y \right)}\, dx = \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(y \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2} + C$$$A