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Beliebt Rechnen >

integral of x(cos(x))^3

Lösung

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

Lösung

x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

Schritte zur Lösung

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

Vereinfache

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

Wende die partielle Integration an

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

\int sin (x) - \frac{1}{3} sin^{3} (x) d x = - \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)

= x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x))

Vereinfache

x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)) : x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

Füge eine Konstante zur Lösung hinzu

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

Graph

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