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Beliebt Rechnen >

integral of cos^3(x)cos(3x)

Lösung

\int cos^{3} (x) cos (3 x) d x

Lösung

\frac{1}{2} cos^{2} (x) (\frac{1}{4} sin (4 x) + \frac{1}{2} sin (2 x)) + \frac{1}{2} (\frac{1}{8} (\frac{1}{2} sin (2 x) - \frac{1}{6} sin (6 x)) + \frac{1}{4} (x - \frac{1}{4} sin (4 x))) + C

Schritte zur Lösung

\int cos^{3} (x) cos (3 x) d x

Vereinfache

cos^{3} (x) : cos^{2} (x) cos (x)

= \int cos^{2} (x) cos (x) cos (3 x) d x

Wende die partielle Integration an

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

\int - \frac{1}{2} sin (2 x) (\frac{1}{4} sin (4 x) + \frac{1}{2} sin (2 x)) d x = - \frac{1}{2} (\frac{1}{8} (\frac{1}{2} sin (2 x) - \frac{1}{6} sin (6 x)) + \frac{1}{4} (x - \frac{1}{4} sin (4 x)))

= \frac{1}{2} cos^{2} (x) (\frac{1}{4} sin (4 x) + \frac{1}{2} sin (2 x)) - (- \frac{1}{2} (\frac{1}{8} (\frac{1}{2} sin (2 x) - \frac{1}{6} sin (6 x)) + \frac{1}{4} (x - \frac{1}{4} sin (4 x))))

Vereinfache

= \frac{1}{2} cos^{2} (x) (\frac{1}{4} sin (4 x) + \frac{1}{2} sin (2 x)) + \frac{1}{2} (\frac{1}{8} (\frac{1}{2} sin (2 x) - \frac{1}{6} sin (6 x)) + \frac{1}{4} (x - \frac{1}{4} sin (4 x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} cos^{2} (x) (\frac{1}{4} sin (4 x) + \frac{1}{2} sin (2 x)) + \frac{1}{2} (\frac{1}{8} (\frac{1}{2} sin (2 x) - \frac{1}{6} sin (6 x)) + \frac{1}{4} (x - \frac{1}{4} sin (4 x))) + C

Graph

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