7.1 Integration By Parts/10

$f'(x)=\int _{}^{}\sin ^{-1}(x)\cdot dx$

$\int _{}^{}\sin ^{-1}(x)dx$ = $x\sin ^{-1}(x)-\int _{}^{}{\frac {x}{\sqrt {1-x^{2}}}}dx$ = $x\sin ^{-1}(x)+{\frac {1}{2}}\int _{}^{}{\frac {1}{\sqrt {u}}}$ = $x\sin ^{-1}(x)+{\frac {1}{2}}\int _{}^{}u^{-{\frac {1}{2}}}du$ = ${\frac {\pi }{6}}-{\frac {1}{2}}\int _{1}^{\frac {3}{4}}u^{-{\frac {1}{2}}}du$