English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cos(aln(x/b))

Lösung

\int cos (a ln (\frac{x}{b})) d x

\frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b})) + C

Schritte zur Lösung

\int cos (a ln (\frac{x}{b})) d x

Wende U-Substitution an

= \int \frac{e ^{\frac{u}{a}} b cos ( u )}{a} d u

Entferne die Konstante:

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

= \frac{b}{a} \cdot \int e^{\frac{u}{a}} cos (u) d u

Wende U-Substitution an

= \frac{b}{a} \cdot \int e^{v} cos (v a) a d v

Entferne die Konstante:

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

= \frac{b}{a} a \cdot \int e^{v} cos (v a) d v

Wende die partielle Integration an

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \int e^{v} \frac{1}{a} sin (v a) d v)

Entferne die Konstante:

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

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} \cdot \int e^{v} sin (v a) d v)

Wende die partielle Integration an

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - \int - e^{v} \frac{1}{a} cos (a v) d v))

Entferne die Konstante:

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

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - (- \frac{1}{a} \cdot \int e^{v} cos (a v) d v)))

Deshalb

\frac{b}{a} \cdot \int e^{v} cos (a v) d v a = \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - (- \frac{1}{a} \cdot \int e^{v} cos (a v) d v)))

Isoliere

\int e^{v} cos (a v) d v

= \frac{b}{a} (\frac{a e ^{v} sin ( a v )}{a ^{2} + 1} + \frac{e ^{v} cos ( a v )}{a ^{2} + 1}) a

Ersetze zurück

= \frac{b}{a} \frac{a e ^{\frac{a l n ( \frac{x}{b} )}{a}} sin ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} + \frac{e ^{\frac{a l n ( \frac{x}{b} )}{a}} cos ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} a

Vereinfache

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

= \frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b}))

Füge eine Konstante zur Lösung hinzu

= \frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b})) + C

t an g e n t f (x) = \frac{1}{x - 2}

\frac{d}{d x} (3 1 + tan (x))

y^{^{'}} - 2 x (y^{2} + 1) = 0

\frac{d}{d x} ((4 x + 1)^{2} (1 - x)^{3})

x \to 0 lim (\frac{x + 1}{x + 4})