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$$$- \frac{1}{\sqrt{1 - x^{2}}}$$$ 的積分
您的輸入
求$$$\int \left(- \frac{1}{\sqrt{1 - x^{2}}}\right)\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-1$$$ 與 $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$：
$${\color{red}{\int{\left(- \frac{1}{\sqrt{1 - x^{2}}}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{1 - x^{2}}} d x}\right)}}$$
$$$\frac{1}{\sqrt{1 - x^{2}}}$$$ 的積分是 $$$\int{\frac{1}{\sqrt{1 - x^{2}}} d x} = \operatorname{asin}{\left(x \right)}$$$：
$$- {\color{red}{\int{\frac{1}{\sqrt{1 - x^{2}}} d x}}} = - {\color{red}{\operatorname{asin}{\left(x \right)}}}$$
因此，
$$\int{\left(- \frac{1}{\sqrt{1 - x^{2}}}\right)d x} = - \operatorname{asin}{\left(x \right)}$$
加上積分常數：
$$\int{\left(- \frac{1}{\sqrt{1 - x^{2}}}\right)d x} = - \operatorname{asin}{\left(x \right)}+C$$
答案
$$$\int \left(- \frac{1}{\sqrt{1 - x^{2}}}\right)\, dx = - \operatorname{asin}{\left(x \right)} + C$$$A