integral of sin(ln(5x))

Lösung

\int sin (ln (5 x)) d x

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

Schritte zur Lösung

\int sin (ln (5 x)) d x

Wende U-Substitution an

= \int sin (ln (u)) \frac{1}{5} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int sin (ln (u)) d u

Wende die partielle Integration an

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

Setze in

u = 5 x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (\frac{- x ^{2} + 2}{( x ^{2} + 2 ) ^{2}})

d er i v a t i v e \frac{x ^{\frac{1}{3}} - 4}{x ^{\frac{7}{2}}}

\int \frac{z ( 1 - z )}{( z + 1 ) ( z ^{2} + 1 ) ^{2}} d z

\int_{0}^{t} e^{x} sin (t - x) d x

\frac{d y}{d t} + y = 5 sin (2 t)