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Beliebt Rechnen >

integral from 0 to t of te^{-2t}cos(t)

Lösung

\int_{0}^{t} t e^{- 2 t} cos (t) d t

Lösung

\frac{e ^{- 2 t} t sin ( t ) - 2 e ^{- 2 t} t cos ( t )}{5} - \frac{3}{25} e^{- 2 t} cos (t) + \frac{4}{25} e^{- 2 t} sin (t) + \frac{3}{25}

Schritte zur Lösung

\int_{0}^{t} t e^{- 2 t} cos (t) d t

Wende die partielle Integration an

= [t (\frac{1}{5} e^{- 2 t} sin (t) - \frac{2}{5} e^{- 2 t} cos (t)) - \int \frac{1}{5} (e^{- 2 t} sin (t) - 2 e^{- 2 t} cos (t)) d t]_{0}^{t}

\int \frac{1}{5} (e^{- 2 t} sin (t) - 2 e^{- 2 t} cos (t)) d t = \frac{1}{25} (3 e^{- 2 t} cos (t) - 4 e^{- 2 t} sin (t))

= [t (\frac{1}{5} e^{- 2 t} sin (t) - \frac{2}{5} e^{- 2 t} cos (t)) - \frac{1}{25} (3 e^{- 2 t} cos (t) - 4 e^{- 2 t} sin (t))]_{0}^{t}

Berechne die Grenzen:

t (\frac{1}{5} e^{- 2 t} sin (t) - \frac{2}{5} e^{- 2 t} cos (t)) - \frac{1}{25} (3 e^{- 2 t} cos (t) - 4 e^{- 2 t} sin (t)) + \frac{3}{25}

= t (\frac{1}{5} e^{- 2 t} sin (t) - \frac{2}{5} e^{- 2 t} cos (t)) - \frac{1}{25} (3 e^{- 2 t} cos (t) - 4 e^{- 2 t} sin (t)) + \frac{3}{25}

Vereinfache

= \frac{e ^{- 2 t} t sin ( t ) - 2 e ^{- 2 t} t cos ( t )}{5} - \frac{3}{25} e^{- 2 t} cos (t) + \frac{4}{25} e^{- 2 t} sin (t) + \frac{3}{25}

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integral from 0 to 8 of (8-(y^2)/8)

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