integral of (cos(x)sin(x))^2

Lösung

\int (cos (x) sin (x))^{2} d x

\frac{1}{8} (x - \frac{1}{4} sin (4 x)) + C

Schritte zur Lösung

\int (cos (x) sin (x))^{2} d x

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

= \int cos^{2} (x) sin^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1 - cos ( 4 x )}{8} d x

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int 1 - cos (4 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}{8} (\int 1 d x - \int cos (4 x) d x)

\int 1 d x = x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= \frac{1}{8} (x - \frac{1}{4} sin (4 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (x - \frac{1}{4} sin (4 x)) + C

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