integral of (x-sin(x))^2sin(x)

Lösung

\int (x - sin (x))^{2} sin (x) d x

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

Schritte zur Lösung

\int (x - sin (x))^{2} sin (x) d x

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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