5.5 The Substitution Rule/41

$\int {\frac {1}{{\sqrt {1-x^{2}}}\arcsin {x}}}$

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

${\begin{aligned}\int ({\frac {1}{\sqrt {1-x^{2}}}}{\frac {1}{\arcsin {x}}})\;dx&=\int {\frac {1}{u}}\;du\\[2ex]&=\ln |u|+c\\[2ex]&=\ln |\arcsin {x}|+c\\[2ex]\end{aligned}}$