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Beliebt Rechnen >

integral of sin(wx+pi/3)sin(wx)

Lösung

\int sin (w x + \frac{π}{3}) sin (w x) d x

Lösung

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

Schritte zur Lösung

\int sin (w x + \frac{π}{3}) sin (w x) d x

Wende U-Substitution an

= \int \frac{sin ( u ) ( \frac{3}{2} cos ( u ) + \frac{1}{2} sin ( u ) )}{w} d u

Entferne die Konstante:

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

= \frac{1}{w} \cdot \int sin (u) (\frac{3}{2} cos (u) + \frac{1}{2} sin (u)) d u

Multipliziere aus

sin (u) (\frac{3}{2} cos (u) + \frac{1}{2} sin (u)) : \frac{3}{2} sin (u) cos (u) + \frac{1}{2} sin^{2} (u)

= \frac{1}{w} \cdot \int \frac{3}{2} sin (u) cos (u) + \frac{1}{2} sin^{2} (u) d u

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}{w} (\int \frac{3}{2} sin (u) cos (u) d u + \int \frac{1}{2} sin^{2} (u) d u)

\int \frac{3}{2} sin (u) cos (u) d u = \frac{3}{4} sin^{2} (u)

\int \frac{1}{2} sin^{2} (u) d u = \frac{1}{4} (u - \frac{1}{2} sin (2 u))

= \frac{1}{w} (\frac{3}{4} sin^{2} (u) + \frac{1}{4} (u - \frac{1}{2} sin (2 u)))

Setze in

u = w x

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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