integral of (x^2sin(x)cos(x)+xsin^2(x))

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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