7.1 Integration By Parts/27

$f'(x)=\int _{0}^{\frac {1}{2}}\cos ^{-1}(x)\cdot dx$

$\int _{0}^{\frac {1}{2}}\cos ^{-1}(x)dx$ = Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\cos ^{-1}(x){\bigg |}_{0}^{\frac {1}{2}}+\int _{0}^{\frac {1}{2}}{\frac {x}{\sqrt {1-x^{2}}}}dx} = $({\frac {1}{2}}\cdot {\frac {\pi }{3}}){-{\frac {1}{2}}}\int _{1}^{\frac {3}{4}}{\frac {1}{\sqrt {u}}}du$ = ${\frac {\pi }{6}}-{\frac {1}{2}}\int _{1}^{\frac {3}{4}}u^{-{\frac {1}{2}}}du$

= Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\pi }{6}}-{({\frac {1}{2}}}(2u^{\frac {1}{2}})){\bigg |}_{1}^{\frac {3}{4}}} = ${\frac {\pi }{6}}-{\frac {1}{2}}({\sqrt {3}}-1)$ = Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\pi }{6}}-{\frac {\sqrt {3}}{2}}+1} = ${\frac {1}{6}}(\pi +6-3{\sqrt {3}})$

${u}$ = Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\cos ^{-1}(x)}}

${du}$ = ${-sint}dt$

${-du}$ = ${sint}dt$

${u}$ = ${u}$ , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {dv}} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{u}du}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {du}} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {du}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {v}} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{u}}