integral of (4x^3+9x-7)*cos(4x)

Lösung

\int (4 x^{3} + 9 x - 7) \cdot cos (4 x) d x

x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) + \frac{15}{8} x sin (4 x) - \frac{7}{4} sin (4 x) + \frac{15}{32} cos (4 x) + C

Schritte zur Lösung

\int (4 x^{3} + 9 x - 7) cos (4 x) d x

Multipliziere aus

(4 x^{3} + 9 x - 7) cos (4 x) : 4 x^{3} cos (4 x) + 9 x cos (4 x) - 7 cos (4 x)

= \int 4 x^{3} cos (4 x) + 9 x cos (4 x) - 7 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 4 x^{3} cos (4 x) d x + \int 9 x cos (4 x) d x - \int 7 cos (4 x) d x

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

\int 9 x cos (4 x) d x = \frac{9}{16} (4 x sin (4 x) + cos (4 x))

\int 7 cos (4 x) d x = \frac{7}{4} sin (4 x)

= x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) - \frac{3}{8} x sin (4 x) - \frac{3}{32} cos (4 x) + \frac{9}{16} (4 x sin (4 x) + cos (4 x)) - \frac{7}{4} sin (4 x)

Vereinfache

x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) - \frac{3}{8} x sin (4 x) - \frac{3}{32} cos (4 x) + \frac{9}{16} (4 x sin (4 x) + cos (4 x)) - \frac{7}{4} sin (4 x) : x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) + \frac{15}{8} x sin (4 x) - \frac{7}{4} sin (4 x) + \frac{15}{32} cos (4 x)

= x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) + \frac{15}{8} x sin (4 x) - \frac{7}{4} sin (4 x) + \frac{15}{32} cos (4 x)

Füge eine Konstante zur Lösung hinzu

= x^{3} sin (4 x) + \frac{3}{4} x^{2} cos (4 x) + \frac{15}{8} x sin (4 x) - \frac{7}{4} sin (4 x) + \frac{15}{32} cos (4 x) + C

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