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Beliebt Rechnen >

integral from 0 to 6pi of x^2sin(2x)

Lösung

\int_{0}^{6 π} x^{2} sin (2 x) d x

Lösung

- 18 π^{2}

+1

Dezimale

- 177.65287 \dots

Schritte zur Lösung

\int_{0}^{6 π} x^{2} sin (2 x) d x

Wende die partielle Integration an

= [\frac{1}{2} (- x^{2} cos (2 x) - 2 \cdot \int - x cos (2 x) d x)]_{0}^{6 π}

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

= [\frac{1}{2} (- x^{2} cos (2 x) - 2 (- \frac{1}{4} (2 x sin (2 x) + cos (2 x))))]_{0}^{6 π}

Vereinfache

[\frac{1}{2} (- x^{2} cos (2 x) - 2 (- \frac{1}{4} (2 x sin (2 x) + cos (2 x))))]_{0}^{6 π} : [\frac{1}{4} (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))]_{0}^{6 π}

= [\frac{1}{4} (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))]_{0}^{6 π}

Berechne die Grenzen:

\frac{- 72 π ^{2} + 1}{4} - \frac{1}{4}

= \frac{- 72 π ^{2} + 1}{4} - \frac{1}{4}

Vereinfache

= - 18 π^{2}

Graph

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