integral of (x^2+2x+1)sin(3x)

Lösung

\int (x^{2} + 2 x + 1) sin (3 x) d x

- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{2}{3} x cos (3 x) + \frac{2}{9} sin (3 x) - \frac{7}{27} cos (3 x) + C

Schritte zur Lösung

\int (x^{2} + 2 x + 1) sin (3 x) d x

Multipliziere aus

(x^{2} + 2 x + 1) sin (3 x) : x^{2} sin (3 x) + 2 x sin (3 x) + sin (3 x)

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

\int x^{2} sin (3 x) d x = - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))

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

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

= - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)) + 2 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) - \frac{1}{3} cos (3 x)

Vereinfache

- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)) + 2 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) - \frac{1}{3} cos (3 x) : - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{2}{3} x cos (3 x) + \frac{2}{9} sin (3 x) - \frac{7}{27} cos (3 x)

= - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{2}{3} x cos (3 x) + \frac{2}{9} sin (3 x) - \frac{7}{27} cos (3 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{2}{3} x cos (3 x) + \frac{2}{9} sin (3 x) - \frac{7}{27} cos (3 x) + C

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