integral of x^4sin(9x)

Lösung

\int x^{4} sin (9 x) d x

- \frac{1}{9} x^{4} cos (9 x) + \frac{4}{9} (\frac{1}{9} x^{3} sin (9 x) - \frac{1}{3} (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x)))) + C

Schritte zur Lösung

\int x^{4} sin (9 x) d x

Wende die partielle Integration an

= - \frac{1}{9} x^{4} cos (9 x) - \int - \frac{4}{9} x^{3} cos (9 x) d x

\int - \frac{4}{9} x^{3} cos (9 x) d x = - \frac{4}{9} (\frac{1}{9} x^{3} sin (9 x) - \frac{1}{3} (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))))

= - \frac{1}{9} x^{4} cos (9 x) - (- \frac{4}{9} (\frac{1}{9} x^{3} sin (9 x) - \frac{1}{3} (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x)))))

Vereinfache

= - \frac{1}{9} x^{4} cos (9 x) + \frac{4}{9} (\frac{1}{9} x^{3} sin (9 x) - \frac{1}{3} (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{9} x^{4} cos (9 x) + \frac{4}{9} (\frac{1}{9} x^{3} sin (9 x) - \frac{1}{3} (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x)))) + C

d er i v a t i v e [\frac{( x - 1 )}{( x + 3 )}]^{\frac{1}{3}}

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\int_{0}^{1} 1 + \frac{9 x}{4} d x

cos (t) y^{^{'}} + sin (t) y = 2 cos^{3} (t) sin (t) - 1