7.1 Integration By Parts/36: Difference between revisions

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<math>\int_{0}^{\pi}e^{cost}(sin2t)dt</math> = <math>\int_{0}^{\pi}e^{cost}(2sintcost)dt</math> = <math>{-2}\int_{cos0}^{cos\pi}e^{u}{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>\int_{0}^{\pi}e^{cost}(sin2t)dt</math> = <math>\int_{0}^{\pi}e^{cost}(2sintcost)dt</math> = <math>{-2}\int_{cos0}^{cos\pi}e^{u}{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>{u}</math> = <math>(cost)</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>{du}</math> = <math>(-sint)dt</math>


<math>{-du}</math> = <math>(sint)dt</math>
<math>{u}</math> = <math>{cost}</math>


<math>{du}</math> = <math>{-sint}dt</math>


<math>{u}</math> = <math>{u}</math>  <math>{dv}</math> = <math>e^{u}du</math>
<math>{-du}</math> = <math>{sint}dt</math>


<math>{du}</math> = <math>{du}</math>   <math>{v}</math> = <math>e^{u}</math>
 
<math>{u}</math> = <math>{u}</math> ,  <math>{dv}</math> = <math>e^{u}du</math>
 
<math>{du}</math> = <math>{du}</math> ,  <math>{v}</math> = <math>e^{u}</math>

Latest revision as of 03:07, 23 November 2022

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