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

Lösung

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

- \frac{2}{3} x^{2} cos (3 x) + \frac{4}{9} x sin (3 x) - x cos (3 x) + \frac{1}{3} sin (3 x) - \frac{14}{27} cos (3 x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

2 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) - x cos (3 x) + \frac{1}{3} sin (3 x) - \frac{2}{3} cos (3 x) : - \frac{2}{3} x^{2} cos (3 x) + \frac{4}{9} x sin (3 x) - x cos (3 x) + \frac{1}{3} sin (3 x) - \frac{14}{27} cos (3 x)

= - \frac{2}{3} x^{2} cos (3 x) + \frac{4}{9} x sin (3 x) - x cos (3 x) + \frac{1}{3} sin (3 x) - \frac{14}{27} cos (3 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{3} x^{2} cos (3 x) + \frac{4}{9} x sin (3 x) - x cos (3 x) + \frac{1}{3} sin (3 x) - \frac{14}{27} cos (3 x) + C