integral of e^{6x}sin(5x)

Lösung

\int e^{6 x} sin (5 x) d x

- \frac{5 e ^{6 x} cos ( 5 x )}{61} + \frac{6 e ^{6 x} sin ( 5 x )}{61} + C

Schritte zur Lösung

\int e^{6 x} sin (5 x) d x

Wende die partielle Integration an

= - \frac{1}{5} e^{6 x} cos (5 x) - \int - \frac{6}{5} e^{6 x} cos (5 x) d x

Entferne die Konstante:

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

= - \frac{1}{5} e^{6 x} cos (5 x) - (- \frac{6}{5} \cdot \int e^{6 x} cos (5 x) d x)

Wende die partielle Integration an

= - \frac{1}{5} e^{6 x} cos (5 x) - (- \frac{6}{5} (\frac{1}{5} e^{6 x} sin (5 x) - \int \frac{6}{5} e^{6 x} sin (5 x) d x))

Entferne die Konstante:

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

= - \frac{1}{5} e^{6 x} cos (5 x) - (- \frac{6}{5} (\frac{1}{5} e^{6 x} sin (5 x) - \frac{6}{5} \cdot \int e^{6 x} sin (5 x) d x))

Deshalb

\int e^{6 x} sin (5 x) d x = - \frac{1}{5} e^{6 x} cos (5 x) - (- \frac{6}{5} (\frac{1}{5} e^{6 x} sin (5 x) - \frac{6}{5} \cdot \int e^{6 x} sin (5 x) d x))

Isoliere

\int e^{6 x} sin (5 x) d x

= - \frac{5 e ^{6 x} cos ( 5 x )}{61} + \frac{6 e ^{6 x} sin ( 5 x )}{61}

Füge eine Konstante zur Lösung hinzu

= - \frac{5 e ^{6 x} cos ( 5 x )}{61} + \frac{6 e ^{6 x} sin ( 5 x )}{61} + C

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