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Beliebt Rechnen >

integral of sin(4x)(1-cos(4x))^3

Lösung

\int sin (4 x) (1 - cos (4 x))^{3} d x

Lösung

- \frac{1}{4} cos (4 x) + \frac{3}{8} cos^{2} (4 x) - \frac{1}{4} cos^{3} (4 x) + \frac{1}{16} cos^{4} (4 x) + C

Schritte zur Lösung

\int sin (4 x) (1 - cos (4 x))^{3} d x

Multipliziere aus

sin (4 x) (1 - cos (4 x))^{3} : sin (4 x) - 3 sin (4 x) cos (4 x) + 3 cos^{2} (4 x) sin (4 x) - cos^{3} (4 x) sin (4 x)

= \int sin (4 x) - 3 sin (4 x) cos (4 x) + 3 cos^{2} (4 x) sin (4 x) - cos^{3} (4 x) sin (4 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

= \int sin (4 x) d x - \int 3 sin (4 x) cos (4 x) d x + \int 3 cos^{2} (4 x) sin (4 x) d x - \int cos^{3} (4 x) sin (4 x) d x

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

\int 3 sin (4 x) cos (4 x) d x = - \frac{3}{8} cos^{2} (4 x)

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

\int cos^{3} (4 x) sin (4 x) d x = - \frac{1}{16} cos^{4} (4 x)

= - \frac{1}{4} cos (4 x) - (- \frac{3}{8} cos^{2} (4 x)) - \frac{1}{4} cos^{3} (4 x) - (- \frac{1}{16} cos^{4} (4 x))

Vereinfache

= - \frac{1}{4} cos (4 x) + \frac{3}{8} cos^{2} (4 x) - \frac{1}{4} cos^{3} (4 x) + \frac{1}{16} cos^{4} (4 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} cos (4 x) + \frac{3}{8} cos^{2} (4 x) - \frac{1}{4} cos^{3} (4 x) + \frac{1}{16} cos^{4} (4 x) + C

Graph

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