integral from 0 to a of sin(cx)sin(cx)

Lösung

\int_{0}^{a} sin (c x) sin (c x) d x

\frac{1}{2} a - \frac{1}{4 c} sin (2 c a)

Schritte zur Lösung

\int_{0}^{a} sin (c x) sin (c x) d x

Vereinfache

sin (c x) sin (c x) : sin^{2} (c x)

= \int_{0}^{a} sin^{2} (c x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{a} 1 - cos^{2} (c 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

= \int_{0}^{a} 1 d x - \int_{0}^{a} cos^{2} (c x) d x

\int_{0}^{a} 1 d x = a

\int_{0}^{a} cos^{2} (c x) d x = \frac{1}{2} (a + \frac{sin ( 2 c a )}{2 c})

= a - \frac{1}{2} (a + \frac{sin ( 2 c a )}{2 c})

Vereinfache

a - \frac{1}{2} (a + \frac{sin ( 2 c a )}{2 c}) : \frac{1}{2} a - \frac{1}{4 c} sin (2 c a)

= \frac{1}{2} a - \frac{1}{4 c} sin (2 c a)

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