integral of x^2sin^2(ax)

Lösung

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

\frac{1}{2} x^{3} - \frac{1}{4 a} x^{2} sin (2 a x) - \frac{x ^{3}}{3} - \frac{1}{4 a ^{2}} x cos (2 a x) + \frac{1}{8 a ^{3}} sin (2 a x) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int x^{2} - \frac{1}{2 a} x sin (2 a x) d x = \frac{x ^{3}}{3} - \frac{1}{8 a ^{3}} (- 2 a x cos (2 a x) + sin (2 a x))

= x^{2} (\frac{1}{2} x - \frac{1}{4 a} sin (2 a x)) - (\frac{x ^{3}}{3} - \frac{1}{8 a ^{3}} (- 2 a x cos (2 a x) + sin (2 a x)))

Vereinfache

x^{2} (\frac{1}{2} x - \frac{1}{4 a} sin (2 a x)) - (\frac{x ^{3}}{3} - \frac{1}{8 a ^{3}} (- 2 a x cos (2 a x) + sin (2 a x))) : \frac{1}{2} x^{3} - \frac{1}{4 a} x^{2} sin (2 a x) - \frac{x ^{3}}{3} - \frac{1}{4 a ^{2}} x cos (2 a x) + \frac{1}{8 a ^{3}} sin (2 a x)

= \frac{1}{2} x^{3} - \frac{1}{4 a} x^{2} sin (2 a x) - \frac{x ^{3}}{3} - \frac{1}{4 a ^{2}} x cos (2 a x) + \frac{1}{8 a ^{3}} sin (2 a x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{3} - \frac{1}{4 a} x^{2} sin (2 a x) - \frac{x ^{3}}{3} - \frac{1}{4 a ^{2}} x cos (2 a x) + \frac{1}{8 a ^{3}} sin (2 a x) + C

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