de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln ∣ x + 1 ∣

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

(t^2+16)^2y^'+2t(t^2+16)y=4,y(-2)=10

(t^{2} + 16)^{2} y^{^{'}} + 2 t (t^{2} + 16) y = 4, y (- 2) = 10

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

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

integral of sqrt(36-4x^2)

\int 36 - 4 x^{2} d x

integral of 3xe^{-3x}

\int 3 x e^{- 3 x} d x

integral of e^{-x}sin(4x)

\int e^{- x} sin (4 x) d x