integral of sin(x)-1/3 sin^3(x)

Lösung

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

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

Schritte zur Lösung

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

\int sin (x) d x = - cos (x)

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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