integral of e^{i8x}cos(3x)

Lösung

\int e^{i 8 x} cos (3 x) d x

\frac{1}{165} (- 20 sin (11 x) + 44 sin (5 x) + 55 cos (8 x) sin (3 x)) + i \frac{1}{165} (20 cos (11 x) - 44 cos (5 x) + 55 sin (3 x) sin (8 x)) + C

Schritte zur Lösung

\int e^{i 8 x} cos (3 x) d x

Wende die partielle Integration an

= \frac{1}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \int \frac{1}{3} sin (3 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}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \frac{1}{3} \cdot \int sin (3 x) (- 8 sin (8 x) + 8 i cos (8 x)) d x

Schreibe

sin (3 x) (- 8 sin (8 x) + 8 i cos (8 x))

in der Standard komplexen Form um:

- 8 sin (3 x) sin (8 x) + 8 sin (3 x) cos (8 x) i

= \frac{1}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \frac{1}{3} \cdot \int - 8 sin (3 x) sin (8 x) + 8 i sin (3 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}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \frac{1}{3} (- \int 8 sin (3 x) sin (8 x) d x + \int 8 i sin (3 x) cos (8 x) d x)

\int 8 sin (3 x) sin (8 x) d x = 4 (\frac{1}{5} sin (5 x) - \frac{1}{11} sin (11 x))

\int 8 i sin (3 x) cos (8 x) d x = i \frac{1}{55} (- 20 cos (11 x) + 44 cos (5 x))

= \frac{1}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \frac{1}{3} (- 4 (\frac{1}{5} sin (5 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{55} (- 20 cos (11 x) + 44 cos (5 x)))

Vereinfache

\frac{1}{3} sin (3 x) (cos (8 x) + i sin (8 x)) - \frac{1}{3} (- 4 (\frac{1}{5} sin (5 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{55} (- 20 cos (11 x) + 44 cos (5 x))) : \frac{1}{165} (- 20 sin (11 x) + 44 sin (5 x) + 55 cos (8 x) sin (3 x)) + i \frac{1}{165} (20 cos (11 x) - 44 cos (5 x) + 55 sin (3 x) sin (8 x))

= \frac{1}{165} (- 20 sin (11 x) + 44 sin (5 x) + 55 cos (8 x) sin (3 x)) + i \frac{1}{165} (20 cos (11 x) - 44 cos (5 x) + 55 sin (3 x) sin (8 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{165} (- 20 sin (11 x) + 44 sin (5 x) + 55 cos (8 x) sin (3 x)) + i \frac{1}{165} (20 cos (11 x) - 44 cos (5 x) + 55 sin (3 x) sin (8 x)) + C

\int_{1}^{2} (\frac{x}{2} - \frac{2}{x}) d x

in v erse l a pl a ce \frac{5 e ^{- s}}{( s ^{2} - 2 s + 1 )}

\int \frac{10 x}{4 x ^{2} + 1} d x

\frac{\partial}{\partial x} (x^{8} y^{6} + 3 x^{7} y)

\int (x^{2} x - 5) d x