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$$$\frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}}$$$ 的积分
相关计算器: 定积分与广义积分计算器
您的输入
求$$$\int \frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}}\, dx$$$。
解答
设$$$u=\operatorname{asin}{\left(x \right)}$$$。
则$$$du=\left(\operatorname{asin}{\left(x \right)}\right)^{\prime }dx = \frac{dx}{\sqrt{1 - x^{2}}}$$$ (步骤见»)，并有$$$\frac{dx}{\sqrt{1 - x^{2}}} = du$$$。
该积分可以改写为
$${\color{red}{\int{\frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$
$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:
$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$
回忆一下 $$$u=\operatorname{asin}{\left(x \right)}$$$:
$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\operatorname{asin}{\left(x \right)}}}}\right| \right)}$$
因此，
$$\int{\frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}} d x} = \ln{\left(\left|{\operatorname{asin}{\left(x \right)}}\right| \right)}$$
加上积分常数：
$$\int{\frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}} d x} = \ln{\left(\left|{\operatorname{asin}{\left(x \right)}}\right| \right)}+C$$
答案
$$$\int \frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}}\, dx = \ln\left(\left|{\operatorname{asin}{\left(x \right)}}\right|\right) + C$$$A