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https://texercises.com/exercise/trigonometrisches-integral/
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Exercise:
Zeige: _^picos^thetashetarm dtheta frac Hinweis: Verwe Partielle Integration.

Solution:
* I _^picos^thetashetarm dtheta * Definiere: * fracrm dfthetarm dtheta sheta gtheta cos^theta * Dann sind * ftheta -costheta fracrm dgthetarm dtheta -shetacostheta * Für partielle Integration gilt im Allgemeinen: * _a^b fracrm dfthetarm dtheta gtheta leftfgright_a^b - _a^b ftheta fracrm dgthetarm dtheta * In unserem Fall also: * I left-costhetacos^thetaright_^pi-_^pi-costheta-shetacosthetarm dtheta & left-cos^thetaright_^pi-_^picos^thetashetarm dtheta & left-cos^thetaright_^pi-I * Auf beiden Seiten addiert man +I und erhält: I left-cos^thetaright_^pi bzw. * I frac left-cos^thetaright_^pi fracleft-cos^pi+cos^right -left-fracright+frac frac qquad blacksquare *
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\(\LaTeX\)-Code
Exercise:
Zeige: _^picos^thetashetarm dtheta frac Hinweis: Verwe Partielle Integration.

Solution:
* I _^picos^thetashetarm dtheta * Definiere: * fracrm dfthetarm dtheta sheta gtheta cos^theta * Dann sind * ftheta -costheta fracrm dgthetarm dtheta -shetacostheta * Für partielle Integration gilt im Allgemeinen: * _a^b fracrm dfthetarm dtheta gtheta leftfgright_a^b - _a^b ftheta fracrm dgthetarm dtheta * In unserem Fall also: * I left-costhetacos^thetaright_^pi-_^pi-costheta-shetacosthetarm dtheta & left-cos^thetaright_^pi-_^picos^thetashetarm dtheta & left-cos^thetaright_^pi-I * Auf beiden Seiten addiert man +I und erhält: I left-cos^thetaright_^pi bzw. * I frac left-cos^thetaright_^pi fracleft-cos^pi+cos^right -left-fracright+frac frac qquad blacksquare *
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Tags
cosinus, integral, integration, partielle, sinus, trigonometrisches
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Difficulty
(3, default)
Points
3 (default)
Language
GER (Deutsch)
Type
Calculative / Quantity
Creator pw
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