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Beliebt Rechnen >

integral of e^{i7x}cos(4x)

Lösung

\int e^{i 7 x} cos (4 x) d x

Lösung

\frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x)) + C

Schritte zur Lösung

\int e^{i 7 x} cos (4 x) d x

Wende die partielle Integration an

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \int \frac{1}{4} sin (4 x) (- 7 sin (7 x) + 7 i cos (7 x)) d x

Entferne die Konstante:

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

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} \cdot \int sin (4 x) (- 7 sin (7 x) + 7 i cos (7 x)) d x

Schreibe

sin (4 x) (- 7 sin (7 x) + 7 i cos (7 x))

in der Standard komplexen Form um:

- 7 sin (4 x) sin (7 x) + 7 sin (4 x) cos (7 x) i

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} \cdot \int - 7 sin (4 x) sin (7 x) + 7 i sin (4 x) cos (7 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}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \int 7 sin (4 x) sin (7 x) d x + \int 7 i sin (4 x) cos (7 x) d x)

\int 7 sin (4 x) sin (7 x) d x = \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x))

\int 7 i sin (4 x) cos (7 x) d x = i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x))

= \frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x)))

Vereinfache

\frac{1}{4} sin (4 x) (cos (7 x) + i sin (7 x)) - \frac{1}{4} (- \frac{7}{2} (\frac{1}{3} sin (3 x) - \frac{1}{11} sin (11 x)) + i \frac{1}{66} (- 21 cos (11 x) + 77 cos (3 x))) : \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x))

= \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{264} (- 21 sin (11 x) + 77 sin (3 x) + 66 cos (7 x) sin (4 x)) + i \frac{1}{264} (21 cos (11 x) - 77 cos (3 x) + 66 sin (4 x) sin (7 x)) + C

Beliebte Beispiele

integral of e^{2x}x^3

\int e^{2 x} x^{3} d x

inverselaplace ((s-1))/(4s^2+s+24)

in v erse l a pl a ce \frac{( s - 1 )}{4 s ^{2} + s + 24}

derivative of (2x^3-3x^2/((x-1)^2))

\frac{d}{d x} (\frac{2 x ^{3} - 3 x ^{2}}{( x - 1 ) ^{2}})

y^'=(4t^3+1)/(2y-6),y(1)=2

y^{^{'}} = \frac{4 t ^{3} + 1}{2 y - 6}, y (1) = 2

integral of (e^{6/t})/(t^2)

\int \frac{e ^{\frac{6}{t}}}{t ^{2}} d t