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Beliebt Rechnen >

integral of sin^2(ln(x))

Lösung

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

Lösung

\frac{1}{10} (5 x - 2 x sin (2 ln (x)) - x cos (2 ln (x))) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1 - cos ( 2 ln ( x ) )}{2} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int 1 - cos (2 ln (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}{2} (\int 1 d x - \int cos (2 ln (x)) d x)

\int 1 d x = x

\int cos (2 ln (x)) d x = \frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x))

= \frac{1}{2} (x - (\frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x))))

Vereinfache

\frac{1}{2} (x - (\frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x)))) : \frac{1}{10} (5 x - 2 x sin (2 ln (x)) - x cos (2 ln (x)))

= \frac{1}{10} (5 x - 2 x sin (2 ln (x)) - x cos (2 ln (x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} (5 x - 2 x sin (2 ln (x)) - x cos (2 ln (x))) + C

Graph

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