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Beliebt Rechnen >

integral from 0 to t of ue^{-2u}cos(3u)

Lösung

\int_{0}^{t} u e^{- 2 u} cos (3 u) d u

Lösung

\frac{3 t e ^{- 2 t} sin ( 3 t ) - 2 t e ^{- 2 t} cos ( 3 t )}{13} + \frac{5}{169} e^{- 2 t} cos (3 t) + \frac{12}{169} e^{- 2 t} sin (3 t) - \frac{5}{169}

Schritte zur Lösung

\int_{0}^{t} u e^{- 2 u} cos (3 u) d u

Wende die partielle Integration an

= [u (\frac{3}{13} e^{- 2 u} sin (3 u) - \frac{2}{13} e^{- 2 u} cos (3 u)) - \int \frac{1}{13} (3 e^{- 2 u} sin (3 u) - 2 e^{- 2 u} cos (3 u)) d u]_{0}^{t}

\int \frac{1}{13} (3 e^{- 2 u} sin (3 u) - 2 e^{- 2 u} cos (3 u)) d u = \frac{1}{169} (- 5 e^{- 2 u} cos (3 u) - 12 e^{- 2 u} sin (3 u))

= [u (\frac{3}{13} e^{- 2 u} sin (3 u) - \frac{2}{13} e^{- 2 u} cos (3 u)) - \frac{1}{169} (- 5 e^{- 2 u} cos (3 u) - 12 e^{- 2 u} sin (3 u))]_{0}^{t}

Berechne die Grenzen:

t (\frac{3}{13} e^{- 2 t} sin (3 t) - \frac{2}{13} e^{- 2 t} cos (3 t)) - \frac{1}{169} (- 5 e^{- 2 t} cos (3 t) - 12 e^{- 2 t} sin (3 t)) - \frac{5}{169}

= t (\frac{3}{13} e^{- 2 t} sin (3 t) - \frac{2}{13} e^{- 2 t} cos (3 t)) - \frac{1}{169} (- 5 e^{- 2 t} cos (3 t) - 12 e^{- 2 t} sin (3 t)) - \frac{5}{169}

Vereinfache

= \frac{3 t e ^{- 2 t} sin ( 3 t ) - 2 t e ^{- 2 t} cos ( 3 t )}{13} + \frac{5}{169} e^{- 2 t} cos (3 t) + \frac{12}{169} e^{- 2 t} sin (3 t) - \frac{5}{169}

Beliebte Beispiele

integral from 0 to 6 of x^5

\int_{0}^{6} x^{5} d x

integral from-pi to pi of x^2sin^2(x)

\int_{- π}^{π} x^{2} sin^{2} (x) d x

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\int_{- 3}^{0} (x) d x

integral from-2 to 0 of 1/2 x

\int_{- 2}^{0} \frac{1}{2} x d x

integral from 0 to 5/2 of 1/(4x^2+25)

\int_{0}^{\frac{5}{2}} \frac{1}{4 x ^{2} + 25} d x