integral of x^3sin(ax)

Lösung

\int x^{3} sin (a x) d x

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

Schritte zur Lösung

\int x^{3} sin (a x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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