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Beliebt Rechnen >

integral of e^{i8x}cos(5x)

Lösung

\int e^{i 8 x} cos (5 x) d x

Lösung

\frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x)) + C

Schritte zur Lösung

\int e^{i 8 x} cos (5 x) d x

Wende die partielle Integration an

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \int \frac{1}{5} sin (5 x) (- 8 sin (8 x) + 8 i cos (8 x)) d x

Entferne die Konstante:

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

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} \cdot \int sin (5 x) (- 8 sin (8 x) + 8 i cos (8 x)) d x

Schreibe

sin (5 x) (- 8 sin (8 x) + 8 i cos (8 x))

in der Standard komplexen Form um:

- 8 sin (5 x) sin (8 x) + 8 sin (5 x) cos (8 x) i

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} \cdot \int - 8 sin (5 x) sin (8 x) + 8 i sin (5 x) cos (8 x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} (- \int 8 sin (5 x) sin (8 x) d x + \int 8 i sin (5 x) cos (8 x) d x)

\int 8 sin (5 x) sin (8 x) d x = 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x))

\int 8 i sin (5 x) cos (8 x) d x = i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x))

= \frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} (- 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x)) + i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x)))

Vereinfache

\frac{1}{5} sin (5 x) (cos (8 x) + i sin (8 x)) - \frac{1}{5} (- 4 (\frac{1}{3} sin (3 x) - \frac{1}{13} sin (13 x)) + i \frac{1}{39} (- 12 cos (13 x) + 52 cos (3 x))) : \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x))

= \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{195} (- 12 sin (13 x) + 52 sin (3 x) + 39 cos (8 x) sin (5 x)) + i \frac{1}{195} (12 cos (13 x) - 52 cos (3 x) + 39 sin (5 x) sin (8 x)) + C

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