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Beliebt Rechnen >

integral of (x^2cos^2(x))

Lösung

\int (x^{2} cos^{2} (x)) d x

Lösung

\frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x) + C

Schritte zur Lösung

\int x^{2} cos^{2} (x) d x

Wende die partielle Integration an

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

\int - \frac{1}{3} x^{3} sin (2 x) d x = \frac{1}{6} x^{3} cos (2 x) - \frac{1}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

= \frac{1}{3} x^{3} cos^{2} (x) - (\frac{1}{6} x^{3} cos (2 x) - \frac{1}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))

Vereinfache

\frac{1}{3} x^{3} cos^{2} (x) - (\frac{1}{6} x^{3} cos (2 x) - \frac{1}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))) : \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x)

= \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x) + C

Graph

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