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Beliebt Rechnen >

integral of (x^3+2x-1)cos(4x)

Lösung

\int (x^{3} + 2 x - 1) cos (4 x) d x

Lösung

\frac{1}{4} x^{3} sin (4 x) + \frac{3}{16} x^{2} cos (4 x) + \frac{13}{32} x sin (4 x) + \frac{13}{128} cos (4 x) - \frac{1}{4} sin (4 x) + C

Schritte zur Lösung

\int (x^{3} + 2 x - 1) cos (4 x) d x

Multipliziere aus

(x^{3} + 2 x - 1) cos (4 x) : x^{3} cos (4 x) + 2 x cos (4 x) - cos (4 x)

= \int x^{3} cos (4 x) + 2 x cos (4 x) - cos (4 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} cos (4 x) d x + \int 2 x cos (4 x) d x - \int cos (4 x) d x

\int x^{3} cos (4 x) d x = \frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

\int 2 x cos (4 x) d x = \frac{1}{8} (4 x sin (4 x) + cos (4 x))

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= \frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + \frac{1}{8} (4 x sin (4 x) + cos (4 x)) - \frac{1}{4} sin (4 x)

Vereinfache

\frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + \frac{1}{8} (4 x sin (4 x) + cos (4 x)) - \frac{1}{4} sin (4 x) : \frac{1}{4} x^{3} sin (4 x) + \frac{3}{16} x^{2} cos (4 x) + \frac{13}{32} x sin (4 x) + \frac{13}{128} cos (4 x) - \frac{1}{4} sin (4 x)

= \frac{1}{4} x^{3} sin (4 x) + \frac{3}{16} x^{2} cos (4 x) + \frac{13}{32} x sin (4 x) + \frac{13}{128} cos (4 x) - \frac{1}{4} sin (4 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{3} sin (4 x) + \frac{3}{16} x^{2} cos (4 x) + \frac{13}{32} x sin (4 x) + \frac{13}{128} cos (4 x) - \frac{1}{4} sin (4 x) + C

Graph

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