7.1 Integration By Parts/17: Difference between revisions

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(Created page with "<math>\int_{0}^{1}\left(3+x\sqrt{x}\right)dx<math>")
 
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<math>\int_{0}^{1}\left(3+x\sqrt{x}\right)dx<math>
<math>\int\left(e^{2{\theta}}\right)sin{3\theta}  d{\theta}</math>
 
First:
 
<math>u={\sin{(3\theta)}}</math> 
<math>du={3\cos{(3\theta)}} d{\theta}</math> 
<math>dv=\left(e^{2{\theta}}\right)</math>
<math>v=\frac{1}{2}\left(e^{2{\theta}}\right)</math>
 
Then:
<math>I=\int\left(e^{2{\theta}}\right)\sin{3\theta}  d{\theta}=\frac{1}{2}\left(e^{2{\theta}}\right)*{\sin{(3\theta)}}-\frac{3}{2}\int\left(e^{2{\theta}}\right)\cos{3\theta}  d{\theta}</math>
 
Next: let 
 
<math>U={\cos{(3\theta)}}</math>
 
<math>dU={3\sin{(3\theta)}} d{\theta}</math>
 
<math>dV=\left(e^{2{\theta}}\right) d{\theta}</math>
 
<math>V=\frac{1}{2}\left(e^{2{\theta}}\right)</math> to get
 
<math>\int\left(e^{2{\theta}}\right)\cos{3\theta}  d{\theta}=\frac{1}{2}\int\left(e^{2{\theta}}\right)\cos{3\theta}  d{\theta}+\frac{3}{2}\int\left(e^{2{\theta}}\right)\sin{3\theta}  d{\theta}</math> substituting in the previous formula gives
 
<math>I=\frac{1}{2}\left(e^{2{\theta}}\right)\sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)\cos{3\theta}-\frac{9}{4}\int\left(e^{2{\theta}}\right)sin{3\theta}  d{\theta}</math>
 
<math>=\frac{1}{2}\left(e^{2{\theta}}\right)sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)cos{3\theta}-\frac{9}{4}I</math>
 
Then:
<math>\frac{13}{4}I=\frac{1}{2}\left(e^{2{\theta}}\right)sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)cos{3\theta}+C</math>,
 
Hence,<math>I=\frac{1}{13}\left(e^{2{\theta}}\right)2sin{3\theta}-{3cos3\theta}+C</math>,
 
Where
<math>C=\frac{4}{13}C</math>

Latest revision as of 05:23, 6 December 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 \int\left(e^{2{\theta}}\right)sin{3\theta} d{\theta}}

First:

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={\sin{(3\theta)}}} 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={3\cos{(3\theta)}} d{\theta}}

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=\left(e^{2{\theta}}\right)} 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=\frac{1}{2}\left(e^{2{\theta}}\right)}

Then: 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 I=\int\left(e^{2{\theta}}\right)\sin{3\theta} d{\theta}=\frac{1}{2}\left(e^{2{\theta}}\right)*{\sin{(3\theta)}}-\frac{3}{2}\int\left(e^{2{\theta}}\right)\cos{3\theta} d{\theta}}

Next: let

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{(3\theta)}}}

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={3\sin{(3\theta)}} d{\theta}}

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=\left(e^{2{\theta}}\right) d{\theta}}

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=\frac{1}{2}\left(e^{2{\theta}}\right)} to get

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\left(e^{2{\theta}}\right)\cos{3\theta} d{\theta}=\frac{1}{2}\int\left(e^{2{\theta}}\right)\cos{3\theta} d{\theta}+\frac{3}{2}\int\left(e^{2{\theta}}\right)\sin{3\theta} d{\theta}} substituting in the previous formula gives

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 I=\frac{1}{2}\left(e^{2{\theta}}\right)\sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)\cos{3\theta}-\frac{9}{4}\int\left(e^{2{\theta}}\right)sin{3\theta} d{\theta}}

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}\left(e^{2{\theta}}\right)sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)cos{3\theta}-\frac{9}{4}I}

Then: 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{13}{4}I=\frac{1}{2}\left(e^{2{\theta}}\right)sin{3\theta}-\frac{3}{4}\left(e^{2{\theta}}\right)cos{3\theta}+C} ,

Hence,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 I=\frac{1}{13}\left(e^{2{\theta}}\right)2sin{3\theta}-{3cos3\theta}+C} ,

Where 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 C=\frac{4}{13}C}

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