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