7.1 Integration By Parts/27: Difference between revisions

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<math> f'(x)= \int_{0}^{\frac{1}{2}}\cos^{-1}(x)\cdot dx </math> <br><br>
<math> f'(x)= \int_{0}^{\frac{1}{2}}\cos^{-1}(x)\cdot dx </math> <br><br>
<math>\int_{0}^{\frac{1}{2}}\cos^{-1}(x)dx</math> = <math>x\cos^{-1}(x)\bigg|_{0}^{\frac{1}{2}}+\int_{0}^{\frac{1}{2}}\frac{x}{\sqrt{1-x^2}}dx</math> = <math>\frac{1}{2}\cdot\frac{\pi}{3}{-\frac{1}{2}}\int_{1}^{\frac{3}{4}}\frac{1}{\sqrt{u}}du</math> = <math>{-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}-(-2)\int_{cos0}^{cos\pi}e^{u}du</math>  
<math>\int_{0}^{\frac{1}{2}}\cos^{-1}(x)dx</math> = <math>x\cos^{-1}(x)\bigg|_{0}^{\frac{1}{2}}+\int_{0}^{\frac{1}{2}}\frac{x}{\sqrt{1-x^2}}dx</math> = <math>(\frac{1}{2}\cdot\frac{\pi}{3}){-\frac{1}{2}}\int_{1}^{\frac{3}{4}}\frac{1}{\sqrt{u}}du</math> = <math>{-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}-(-2)\int_{cos0}^{cos\pi}e^{u}du</math>  


= <math>{-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}+{2}e^{u}\bigg|_{cos0}^{cos\pi} du</math> = <math>{2}{u}e^{u}\bigg|_{cos\pi}^{cos0}-{2}e^{u}\bigg|_{cos\pi}^{cos0}du</math> = <math>{2}{cos(0)}e^{cos(0)}-{2}{cos(\pi)}e^{cos(\pi)}-{2}e^{cos(0)}+{2}e^{cos(\pi)}</math>  
= <math>{-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}+{2}e^{u}\bigg|_{cos0}^{cos\pi} du</math> = <math>{2}{u}e^{u}\bigg|_{cos\pi}^{cos0}-{2}e^{u}\bigg|_{cos\pi}^{cos0}du</math> = <math>{2}{cos(0)}e^{cos(0)}-{2}{cos(\pi)}e^{cos(\pi)}-{2}e^{cos(0)}+{2}e^{cos(\pi)}</math>  


= <math>{2}(1)e^{1}-{2}(-1)e^{-1}-{2}e^{1}+{2}e^{-1}</math> =  <math>{2}e^{1}+{2}e^{-1}-{2}e^{1}+{2}e^{-1}</math> = <math>{2}e^{-1}+{2}e^{-1}</math> = <math> {4}e^{-1} </math>
= <math>{2}(1)e^{1}-{2}(-1)e^{-1}-{2}e^{1}+{2}e^{-1}</math> =  <math>{2}e^{1}+{2}e^{-1}-{2}e^{1}+{2}e^{-1}</math> = <math>{2}e^{-1}+{2}e^{-1}</math> = <math> {4}e^{-1} </math>

Revision as of 23:24, 25 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_{0}^{\frac{1}{2}}\cos^{-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_{0}^{\frac{1}{2}}\cos^{-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\cos^{-1}(x)\bigg|_{0}^{\frac{1}{2}}+\int_{0}^{\frac{1}{2}}\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 (\frac{1}{2}\cdot\frac{\pi}{3}){-\frac{1}{2}}\int_{1}^{\frac{3}{4}}\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 {-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}-(-2)\int_{cos0}^{cos\pi}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 {-2}{u}e^{u}\bigg|_{cos0}^{cos\pi}+{2}e^{u}\bigg|_{cos0}^{cos\pi} 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 {2}{u}e^{u}\bigg|_{cos\pi}^{cos0}-{2}e^{u}\bigg|_{cos\pi}^{cos0}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 {2}{cos(0)}e^{cos(0)}-{2}{cos(\pi)}e^{cos(\pi)}-{2}e^{cos(0)}+{2}e^{cos(\pi)}}

= 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 {2}(1)e^{1}-{2}(-1)e^{-1}-{2}e^{1}+{2}e^{-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 {2}e^{-1}+{2}e^{-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 {4}e^{-1} }

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