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$$$\sqrt{4 - 4 \sin^{2}{\left(x \right)}}$$$ 的積分
您的輸入
求$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx$$$。
解答
簡化被積函數:
$${\color{red}{\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = \sqrt{1 - \sin^{2}{\left(x \right)}}$$$：
$${\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(2 \int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}\right)}}$$
此積分（第二類不完全橢圓積分）不存在閉式表示：
$$2 {\color{red}{\int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = 2 {\color{red}{E\left(x\middle| 1\right)}}$$
因此，
$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)$$
加上積分常數：
$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)+C$$
答案
$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx = 2 E\left(x\middle| 1\right) + C$$$A