integral of 10cos(ln(x))

Lösung

\int 10 cos (ln (x)) d x

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

Schritte zur Lösung

\int 10 cos (ln (x)) d x

Entferne die Konstante:

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

= 10 \cdot \int cos (ln (x)) d x

Wende U-Substitution an

= 10 \cdot \int cos (u) e^{u} d u

Wende die partielle Integration an

= 10 (e^{u} sin (u) - \int e^{u} sin (u) d u)

Wende die partielle Integration an

= 10 (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

= 10 (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Deshalb

10 \cdot \int e^{u} cos (u) d u = 10 (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Isoliere

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

= 10 (\frac{e ^{u} sin ( u )}{2} + \frac{e ^{u} cos ( u )}{2})

Setze in

u = ln (x)

ein

= 10 (\frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2})

Vereinfache

10 (\frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2}) : 10 (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)))

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{4 x}{x ^{2} + 3 x + 2} d x

\int \frac{4 x ^{3} - 6 x ^{2} - 7 x}{x ^{2}} d x

\int e u d u

\int cos (x) \cdot e^{- x} d x

\int (\frac{10}{x ^{2}}) d x