7.1 Integration By Parts/10: Difference between revisions

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<math>\int_{}^{}\sin^{-1}(x)dx</math> = <math>x\sin^{-1}(x)-\int_{}^{}\frac{x}{\sqrt{1-x^2}}dx</math> = <math>x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}\frac{1}{\sqrt{u}}du</math> = <math>x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}u^{-\frac{1}{2}}du</math> = <math>x\sin^{-1}(x)+\frac{1}{2}(2u^{\frac{1}{2}})</math>  
<math>\int_{}^{}\sin^{-1}(x)dx</math> = <math>x\sin^{-1}(x)-\int_{}^{}\frac{x}{\sqrt{1-x^2}}dx</math> = <math>x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}\frac{1}{\sqrt{u}}du</math> = <math>x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}u^{-\frac{1}{2}}du</math> = <math>x\sin^{-1}(x)+\frac{1}{2}(2u^{\frac{1}{2}})</math>  
= <math>x\sin^{-1}(x)+\sqrt{u}</math> = <math>x\sin^{-1}(x)+\sqrt{1-x^2}+C</math>
= <math>x\sin^{-1}(x)+\sqrt{u}</math> = <math>x\sin^{-1}(x)+\sqrt{1-x^2}+C</math>
<math>{u}</math> = <math>{1-x^2}</math>
<math>{du}</math> = <math>{-2x}</math>
<math>{-\frac{1}{2}du}</math> = <math>{x}dx</math>
<math>{u}</math> = <math>{\sin^{-1}(x)}</math> ,  <math>{dv}</math> = <math>dx</math>
<math>{du}</math> = <math>{\frac{1}{\sqrt{1-x^2}}dx}</math> ,  <math>{v}</math> = <math>{x}</math>

Latest revision as of 01:49, 27 November 2022

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 f'(x)= \int_{}^{}\sin^{-1}(x)\cdot 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 \int_{}^{}\sin^{-1}(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 x\sin^{-1}(x)-\int_{}^{}\frac{x}{\sqrt{1-x^2}}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 x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}\frac{1}{\sqrt{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 x\sin^{-1}(x)+\frac{1}{2}\int_{}^{}u^{-\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\sin^{-1}(x)+\frac{1}{2}(2u^{\frac{1}{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 x\sin^{-1}(x)+\sqrt{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 x\sin^{-1}(x)+\sqrt{1-x^2}+C}


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}} = 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 {\sin^{-1}(x)}} , 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 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 {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 {\frac{1}{\sqrt{1-x^2}}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 {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 {x}}

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