integral of x^3cos(kx)

Lösung

\int x^{3} cos (k x) d x

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

Schritte zur Lösung

\int x^{3} cos (k x) d x

Wende die partielle Integration an

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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