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$$$\sqrt{4 - 4 \sin^{2}{\left(x \right)}}$$$ 的积分
您的输入
求$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx$$$。
解答
化简被积函数:
$${\color{red}{\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}}$$
对 $$$c=2$$$ 和 $$$f{\left(x \right)} = \sqrt{1 - \sin^{2}{\left(x \right)}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(2 \int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}\right)}}$$
该积分（第二类不完全椭圆积分）没有闭式表达式：
$$2 {\color{red}{\int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = 2 {\color{red}{E\left(x\middle| 1\right)}}$$
因此，
$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)$$
加上积分常数：
$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)+C$$
答案
$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx = 2 E\left(x\middle| 1\right) + C$$$A