5.5 The Substitution Rule/41

$\int {\frac {1}{\sqrt {1-x^{2}}}}=\int {\frac {1}{u}}du=\ln |u|+c=\ln |\arcsin {x}|+c$

${\begin{aligned}u&=\arcsin {x}\\[2ex]du&={\frac {1}{2}}\ {\frac {1}{\sqrt {t}}}dx\\[2ex]2du&={\frac {1}{\sqrt {t}}}dx\end{aligned}}$