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Beliebt Rechnen >

integral of (x^3+5x^2)sin(3x)

Lösung

\int (x^{3} + 5 x^{2}) sin (3 x) d x

Lösung

- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + 5 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) + C

Schritte zur Lösung

\int (x^{3} + 5 x^{2}) sin (3 x) d x

Multipliziere aus

(x^{3} + 5 x^{2}) sin (3 x) : x^{3} sin (3 x) + 5 x^{2} sin (3 x)

= \int x^{3} sin (3 x) + 5 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 x^{3} sin (3 x) d x + \int 5 x^{2} sin (3 x) d x

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

\int 5 x^{2} sin (3 x) d x = 5 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)))

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + 5 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + 5 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) + C

Graph

Beliebte Beispiele

integral of 5xe^{-15x}

\int 5 x e^{- 15 x} d x

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\int \frac{sinh ( x )}{( 1 + cosh ( x ) ) ^{3}} d x

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

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\int \frac{1}{9 x + x ln ^{2} ( x )} d x

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\int \frac{( x + 5 )}{( x ^{2} - 6 x - 15 )} d x