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Beliebt Rechnen >

integral of e^{4θ}sin(5θ)

Lösung

\int e^{4 θ} sin (5 θ) d θ

Lösung

- \frac{5 e ^{4 θ} cos ( 5 θ )}{41} + \frac{4 e ^{4 θ} sin ( 5 θ )}{41} + C

Schritte zur Lösung

\int e^{4 θ} sin (5 θ) d θ

Wende die partielle Integration an

= - \frac{1}{5} e^{4 θ} cos (5 θ) - \int - \frac{4}{5} e^{4 θ} cos (5 θ) d θ

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{5} e^{4 θ} cos (5 θ) - (- \frac{4}{5} \cdot \int e^{4 θ} cos (5 θ) d θ)

Wende die partielle Integration an

= - \frac{1}{5} e^{4 θ} cos (5 θ) - (- \frac{4}{5} (\frac{1}{5} e^{4 θ} sin (5 θ) - \int \frac{4}{5} e^{4 θ} sin (5 θ) d θ))

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{5} e^{4 θ} cos (5 θ) - (- \frac{4}{5} (\frac{1}{5} e^{4 θ} sin (5 θ) - \frac{4}{5} \cdot \int e^{4 θ} sin (5 θ) d θ))

Deshalb

\int e^{4 θ} sin (5 θ) d θ = - \frac{1}{5} e^{4 θ} cos (5 θ) - (- \frac{4}{5} (\frac{1}{5} e^{4 θ} sin (5 θ) - \frac{4}{5} \cdot \int e^{4 θ} sin (5 θ) d θ))

Isoliere

\int e^{4 θ} sin (5 θ) d θ

= - \frac{5 e ^{4 θ} cos ( 5 θ )}{41} + \frac{4 e ^{4 θ} sin ( 5 θ )}{41}

Füge eine Konstante zur Lösung hinzu

= - \frac{5 e ^{4 θ} cos ( 5 θ )}{41} + \frac{4 e ^{4 θ} sin ( 5 θ )}{41} + C

Graph

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