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Exercise:
Solve the following egral: _^pi fraccost-costddt

Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
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Exercise:
Solve the following egral: _^pi fraccost-costddt

Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
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ca, complex, cos, integral, substitution
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(3, default)
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ENG (English)
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Creator rk
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