integral from 0 to x of cos^2(t)

Lösung

\int_{0}^{x} cos^{2} (t) d t

\frac{1}{2} (x + \frac{1}{2} sin (2 x))

Schritte zur Lösung

\int_{0}^{x} cos^{2} (t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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

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

= \frac{1}{2} (x + \frac{1}{2} sin (2 x))

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