integral of-12cos(ln(x))

Lösung

\int - 12 cos (ln (x)) d x

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

Schritte zur Lösung

\int - 12 cos (ln (x)) d x

Entferne die Konstante:

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

= - 12 \cdot \int cos (ln (x)) d x

Wende U-Substitution an

= - 12 \cdot \int cos (u) e^{u} d u

Wende die partielle Integration an

= - 12 (e^{u} sin (u) - \int e^{u} sin (u) d u)

Wende die partielle Integration an

= - 12 (e^{u} sin (u) - (- e^{u} cos (u) - \int - e^{u} cos (u) d u))

Entferne die Konstante:

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

= - 12 (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Deshalb

- 12 \cdot \int e^{u} cos (u) d u = - 12 (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Isoliere

\int e^{u} cos (u) d u

= - 12 (\frac{e ^{u} sin ( u )}{2} + \frac{e ^{u} cos ( u )}{2})

Setze in

u = ln (x)

ein

= - 12 (\frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2})

Vereinfache

- 12 (\frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2}) : - 12 (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)))

= - 12 (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)))

Füge eine Konstante zur Lösung hinzu

= - 12 (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x))) + C

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