∫ 1 1 − x 2 arcsin x {\displaystyle \int {\frac {1}{{\sqrt {1-x^{2}}}\arcsin {x}}}}
u = arcsin x d u = 1 1 − x 2 d x {\displaystyle {\begin{aligned}u&=\arcsin {x}\\[2ex]du&={\frac {1}{\sqrt {1-x^{2}}}}dx\\[2ex]\end{aligned}}}
∫ 1 1 − x 2 arcsin x = ∫ 1 u d u = ln | u | + c = ln | arcsin x | + c {\displaystyle {\begin{aligned}\int {\frac {1}{{\sqrt {1-x^{2}}}\arcsin {x}}}=\int {\frac {1}{u}}du&=\ln |u|+c\\[2ex]&=\ln |\arcsin {x}|+c\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\arcsin {x}\\[2ex]du&={\frac {1}{\sqrt {1-x^{2}}}}dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e5c3841c50e1902bc439001180991f85316c3ab8)
![{\displaystyle {\begin{aligned}\int {\frac {1}{{\sqrt {1-x^{2}}}\arcsin {x}}}=\int {\frac {1}{u}}du&=\ln |u|+c\\[2ex]&=\ln |\arcsin {x}|+c\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e5070826268689372046b438b6cc8cac227d505a)