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Beliebt Rechnen >

integral of usin^2(3u)

Lösung

\int u sin^{2} (3 u) d u

Lösung

\frac{1}{72} (18 u^{2} - 6 u sin (6 u) - cos (6 u)) + C

Schritte zur Lösung

\int u sin^{2} (3 u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1}{2} u (1 - cos (6 u)) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u (1 - cos (6 u)) d u

Wende die partielle Integration an

= \frac{1}{2} (u (u - \frac{1}{6} sin (6 u)) - \int u - \frac{1}{6} sin (6 u) d u)

\int u - \frac{1}{6} sin (6 u) d u = \frac{u ^{2}}{2} + \frac{1}{36} cos (6 u)

= \frac{1}{2} (u (u - \frac{1}{6} sin (6 u)) - (\frac{u ^{2}}{2} + \frac{1}{36} cos (6 u)))

Vereinfache

\frac{1}{2} (u (u - \frac{1}{6} sin (6 u)) - (\frac{u ^{2}}{2} + \frac{1}{36} cos (6 u))) : \frac{1}{72} (18 u^{2} - 6 u sin (6 u) - cos (6 u))

= \frac{1}{72} (18 u^{2} - 6 u sin (6 u) - cos (6 u))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{72} (18 u^{2} - 6 u sin (6 u) - cos (6 u)) + C

Graph

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