integral from 0 to pi of x^3sin(2x)

Lösung

\int_{0}^{π} x^{3} sin (2 x) d x

\frac{- 2 π ^{3} + 3 π}{4}

Dezimale

- 13.14694 \dots

Schritte zur Lösung

\int_{0}^{π} x^{3} sin (2 x) d x

Wende die partielle Integration an

= [\frac{1}{2} (- x^{3} cos (2 x) - 2 \cdot \int - \frac{3}{2} x^{2} cos (2 x) d x)]_{0}^{π}

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

= [\frac{1}{2} (- x^{3} cos (2 x) - 2 (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))))]_{0}^{π}

Vereinfache

[\frac{1}{2} (- x^{3} cos (2 x) - 2 (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))))]_{0}^{π} : [\frac{1}{2} (- x^{3} cos (2 x) + \frac{3}{4} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))]_{0}^{π}

= [\frac{1}{2} (- x^{3} cos (2 x) + \frac{3}{4} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))]_{0}^{π}

Berechne die Grenzen:

\frac{- 2 π ^{3} + 3 π}{4}

= \frac{- 2 π ^{3} + 3 π}{4}

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