integral of cos^2(at)e^{-st}

Lösung

\int cos^{2} (a t) e^{- s t} d t

- e^{- s t} \frac{1}{s} cos^{2} (a t) + 2 a e^{- s t} \frac{a}{s ( 4 a ^{2} + s ^{2} )} cos (2 a t) + e^{- s t} \frac{a}{4 a ^{2} + s ^{2}} sin (2 a t) + C

Schritte zur Lösung

\int cos^{2} (a t) e^{- s t} d t

Wende die partielle Integration an

= - e^{- s t} \frac{1}{s} cos^{2} (a t) - \int a e^{- s t} \frac{1}{s} sin (2 a t) d t

\int a e^{- s t} \frac{1}{s} sin (2 a t) d t = \frac{a}{s ( 4 a ^{2} + s ^{2} )} (- 2 a e^{- s t} cos (2 a t) - s e^{- s t} sin (2 a t))

= - e^{- s t} \frac{1}{s} cos^{2} (a t) - \frac{a}{s ( 4 a ^{2} + s ^{2} )} (- 2 a e^{- s t} cos (2 a t) - s e^{- s t} sin (2 a t))

Vereinfache

- e^{- s t} \frac{1}{s} cos^{2} (a t) - \frac{a}{s ( 4 a ^{2} + s ^{2} )} (- 2 a e^{- s t} cos (2 a t) - s e^{- s t} sin (2 a t)) : - e^{- s t} \frac{1}{s} cos^{2} (a t) + 2 a e^{- s t} \frac{a}{s ( 4 a ^{2} + s ^{2} )} cos (2 a t) + e^{- s t} \frac{a}{4 a ^{2} + s ^{2}} sin (2 a t)

= - e^{- s t} \frac{1}{s} cos^{2} (a t) + 2 a e^{- s t} \frac{a}{s ( 4 a ^{2} + s ^{2} )} cos (2 a t) + e^{- s t} \frac{a}{4 a ^{2} + s ^{2}} sin (2 a t)

Füge eine Konstante zur Lösung hinzu

= - e^{- s t} \frac{1}{s} cos^{2} (a t) + 2 a e^{- s t} \frac{a}{s ( 4 a ^{2} + s ^{2} )} cos (2 a t) + e^{- s t} \frac{a}{4 a ^{2} + s ^{2}} sin (2 a t) + C

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