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Beliebt Rechnen >

integral of (pix-x^2)sin(nx)

Lösung

\int (π x - x^{2}) sin (n x) d x

Lösung

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

Schritte zur Lösung

\int (π x - x^{2}) sin (n x) d x

Multipliziere aus

(π x - x^{2}) sin (n x) : π x sin (n x) - x^{2} sin (n x)

= \int π x sin (n x) - x^{2} sin (n 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 π x sin (n x) d x - \int x^{2} sin (n x) d x

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

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

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

Vereinfache

π (- \frac{1}{n} x cos (n x) + \frac{1}{n ^{2}} sin (n x)) - (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x))) : - \frac{π}{n} x cos (n x) + \frac{π}{n ^{2}} sin (n x) + \frac{1}{n} x^{2} cos (n x) - \frac{2}{n ^{2}} x sin (n x) - \frac{2}{n ^{3}} cos (n x)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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