de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (x-1)/(x^2(x+1)^2)

Lösung

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

Lösung

3 ln ∣ x ∣ + \frac{1}{x} - 3 ln ∣ x + 1 ∣ + \frac{2}{x + 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int \frac{3}{x} - \frac{1}{x ^{2}} - \frac{3}{x + 1} - \frac{2}{( x + 1 ) ^{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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

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

= 3 ln ∣ x ∣ - (- \frac{1}{x}) - 3 ln ∣ x + 1 ∣ - (- \frac{2}{x + 1})

Vereinfache

= 3 ln ∣ x ∣ + \frac{1}{x} - 3 ln ∣ x + 1 ∣ + \frac{2}{x + 1}

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x ∣ + \frac{1}{x} - 3 ln ∣ x + 1 ∣ + \frac{2}{x + 1} + C

Graph

Beliebte Beispiele

tangent f(x)= 7/(x^2)-6x^2,\at x=5

t an g e n t f (x) = \frac{7}{x ^{2}} - 6 x^{2}, at x = 5

derivative of sqrt(2x^2+1)

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

integral of ((1-x))/(e^x)k

\int \frac{( 1 - x )}{e ^{x}} k d x

limit as x approaches 0 of+(ln(x))

x \to 0 lim (+ (ln (x)))

derivative (2x)/5

d er i v a t i v e \frac{2 x}{5}