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Beliebt Rechnen >

integral of sin(ln(8x))

Lösung

\int sin (ln (8 x)) d x

Lösung

\frac{1}{2} x (sin (ln (8 x)) - cos (ln (8 x))) + C

Schritte zur Lösung

\int sin (ln (8 x)) d x

Wende U-Substitution an

= \int sin (ln (u)) \frac{1}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int sin (ln (u)) d u

Wende die partielle Integration an

= \frac{1}{8} (u sin (ln (u)) - \int cos (ln (u)) d u)

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

= \frac{1}{8} (u sin (ln (u)) - (\frac{1}{2} u sin (ln (u)) + \frac{1}{2} u cos (ln (u))))

Setze in

u = 8 x

ein

= \frac{1}{8} (8 x sin (ln (8 x)) - (\frac{1}{2} \cdot 8 x sin (ln (8 x)) + \frac{1}{2} \cdot 8 x cos (ln (8 x))))

Vereinfache

\frac{1}{8} (8 x sin (ln (8 x)) - (\frac{1}{2} \cdot 8 x sin (ln (8 x)) + \frac{1}{2} \cdot 8 x cos (ln (8 x)))) : \frac{1}{2} x (sin (ln (8 x)) - cos (ln (8 x)))

= \frac{1}{2} x (sin (ln (8 x)) - cos (ln (8 x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x (sin (ln (8 x)) - cos (ln (8 x))) + C

Graph

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