integral from 0 to k of sin^2(10pit)

Lösung

\int_{0}^{k} sin^{2} (10 π t) d t

\frac{1}{2} (k - \frac{1}{20 π} sin (20 πk))

Schritte zur Lösung

\int_{0}^{k} sin^{2} (10 π t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{k} \frac{1}{2} (1 - cos (20 π t)) d t

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{k} 1 - cos (20 π t) d t

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}^{k} 1 d t - \int_{0}^{k} cos (20 π t) d t)

\int_{0}^{k} 1 d t = k

\int_{0}^{k} cos (20 π t) d t = \frac{1}{20 π} sin (20 πk)

= \frac{1}{2} (k - \frac{1}{20 π} sin (20 πk))

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