integral of e^{8θ}sin(9θ)

Lösung

\int e^{8 θ} sin (9 θ) d θ

- \frac{9 e ^{8 θ} cos ( 9 θ )}{145} + \frac{8 e ^{8 θ} sin ( 9 θ )}{145} + C

Schritte zur Lösung

\int e^{8 θ} sin (9 θ) d θ

Wende die partielle Integration an

= - \frac{1}{9} e^{8 θ} cos (9 θ) - \int - \frac{8}{9} e^{8 θ} cos (9 θ) d θ

Entferne die Konstante:

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

= - \frac{1}{9} e^{8 θ} cos (9 θ) - (- \frac{8}{9} \cdot \int e^{8 θ} cos (9 θ) d θ)

Wende die partielle Integration an

= - \frac{1}{9} e^{8 θ} cos (9 θ) - (- \frac{8}{9} (\frac{1}{9} e^{8 θ} sin (9 θ) - \int \frac{8}{9} e^{8 θ} sin (9 θ) d θ))

Entferne die Konstante:

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

= - \frac{1}{9} e^{8 θ} cos (9 θ) - (- \frac{8}{9} (\frac{1}{9} e^{8 θ} sin (9 θ) - \frac{8}{9} \cdot \int e^{8 θ} sin (9 θ) d θ))

Deshalb

\int e^{8 θ} sin (9 θ) d θ = - \frac{1}{9} e^{8 θ} cos (9 θ) - (- \frac{8}{9} (\frac{1}{9} e^{8 θ} sin (9 θ) - \frac{8}{9} \cdot \int e^{8 θ} sin (9 θ) d θ))

Isoliere

\int e^{8 θ} sin (9 θ) d θ

= - \frac{9 e ^{8 θ} cos ( 9 θ )}{145} + \frac{8 e ^{8 θ} sin ( 9 θ )}{145}

Füge eine Konstante zur Lösung hinzu

= - \frac{9 e ^{8 θ} cos ( 9 θ )}{145} + \frac{8 e ^{8 θ} sin ( 9 θ )}{145} + C

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