7.1 Integration By Parts/10

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

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

${u}$ = ${1-x^{2}}$

${du}$ = ${-2x}$

${-{\frac {1}{2}}du}$ = ${x}dx$

${u}$ = ${\sin ^{-1}(x)}$ , ${dv}$ = $dx$

${du}$ = ${{\frac {1}{\sqrt {1-x^{2}}}}dx}$ , ${v}$ = ${x}$