de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 4/(x(x-4)^2)

Lösung

\int \frac{4}{x ( x - 4 ) ^{2}} d x

Lösung

\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 4 ∣ - \frac{1}{x - 4} + C

Schritte zur Lösung

\int \frac{4}{x ( x - 4 ) ^{2}} d x

Entferne die Konstante:

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

= 4 \cdot \int \frac{1}{x ( x - 4 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{x ( x - 4 ) ^{2}} : \frac{1}{16 x} - \frac{1}{16 ( x - 4 )} + \frac{1}{4 ( x - 4 ) ^{2}}

= 4 \cdot \int \frac{1}{16 x} - \frac{1}{16 ( x - 4 )} + \frac{1}{4 ( x - 4 ) ^{2}} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{1}{16 x} d x - \int \frac{1}{16 ( x - 4 )} d x + \int \frac{1}{4 ( x - 4 ) ^{2}} d x)

\int \frac{1}{16 x} d x = \frac{1}{16} ln ∣ x ∣

\int \frac{1}{16 ( x - 4 )} d x = \frac{1}{16} ln ∣ x - 4 ∣

\int \frac{1}{4 ( x - 4 ) ^{2}} d x = - \frac{1}{4 ( x - 4 )}

= 4 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )})

Vereinfache

4 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )}) : \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 4 ∣ - \frac{1}{x - 4}

= \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 4 ∣ - \frac{1}{x - 4}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 4 ∣ - \frac{1}{x - 4} + C

Graph

Beliebte Beispiele

limit as x approaches 1+of arcsin(x)

x \to 1 + lim (arcsin (x))

derivative of e^{2x}(3x-2)

\frac{d}{d x} (e^{2 x} (3 x - 2))

erweitern (x^3-4x^2-6x+1)(x^2-3x+1)

e x p an d (x^{3} - 4 x^{2} - 6 x + 1) (x^{2} - 3 x + 1)

derivative of-sin(2x+1)

\frac{d}{d x} (- sin (2 x) + 1)

derivative of (5x-3(4x-3/x))

\frac{d}{d x} ((5 x - 3) (4 x - \frac{3}{x}))