7.1 Integration By Parts/27

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

$\int _{0}^{\frac {1}{2}}\cos ^{-1}(x)dx$ = $x\cos ^{-1}(x){\bigg |}_{0}^{\frac {1}{2}}+\int _{0}^{\frac {1}{2}}{\frac {x}{\sqrt {1-x^{2}}}}dx$ = $({\frac {1}{2}}\cdot {\frac {\pi }{3}}){-{\frac {1}{2}}}\int _{1}^{\frac {3}{4}}{\frac {1}{\sqrt {u}}}du$ = ${\frac {\pi }{6}}-{\frac {1}{2}}\int _{1}^{\frac {3}{4}}u^{-{\frac {1}{2}}}du$

= [change the sign and flip the limits] 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 {\frac{\pi}{6}}+{(\frac{1}{2}}(2u^{\frac{1}{2}}))\bigg|_{\frac{3}{4}}^{1}} = 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 \frac{\pi}{6}+\frac{1}{2}(2-\sqrt{3})} = 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 \frac{\pi}{6}-\frac{\sqrt{3}}{2}+1} = 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 \frac{1}{6}(\pi+6-3\sqrt{3})}

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 {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 {1-x^2}}

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 {-2x}}

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 {-\frac{1}{2}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 {x}dx}

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

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