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Beliebt Rechnen >

integral from 0 to pi of sin(6x)cos(3x)

Lösung

\int_{0}^{π} sin (6 x) cos (3 x) d x

Lösung

\frac{4}{9}

+1

Dezimale

0.44444 \dots

Schritte zur Lösung

\int_{0}^{π} sin (6 x) cos (3 x) d x

Verwende die folgenden Identitäten:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int_{0}^{π} \frac{sin ( 6 x + 3 x ) + sin ( 6 x - 3 x )}{2} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{π} sin (6 x + 3 x) + sin (6 x - 3 x) d x

Vereinfache

= \frac{1}{2} \cdot \int_{0}^{π} sin (9 x) + sin (3 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

= \frac{1}{2} (\int_{0}^{π} sin (9 x) d x + \int_{0}^{π} sin (3 x) d x)

\int_{0}^{π} sin (9 x) d x = \frac{2}{9}

\int_{0}^{π} sin (3 x) d x = \frac{2}{3}

= \frac{1}{2} (\frac{2}{9} + \frac{2}{3})

Vereinfache

\frac{1}{2} (\frac{2}{9} + \frac{2}{3}) : \frac{4}{9}

= \frac{4}{9}

Graph

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