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

Lösung

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

Lösung

- \frac{x ^{2} ln ∣ x - 1 ∣ - x ^{2} ln ∣ x + 1 ∣ - 2 x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣}{8 ( x + 1 ) ( x - 1 )} + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vereinfache

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

= (- \frac{1}{2}) (- \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} + \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )})

Vereinfache

(- \frac{1}{2}) (- \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} + \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )}) : - \frac{x ^{2} ln ∣ x - 1 ∣ - x ^{2} ln ∣ x + 1 ∣ - 2 x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣}{8 ( x + 1 ) ( x - 1 )}

= - \frac{x ^{2} ln ∣ x - 1 ∣ - x ^{2} ln ∣ x + 1 ∣ - 2 x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣}{8 ( x + 1 ) ( x - 1 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{x ^{2} ln ∣ x - 1 ∣ - x ^{2} ln ∣ x + 1 ∣ - 2 x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣}{8 ( x + 1 ) ( x - 1 )} + C

Graph

Beliebte Beispiele

integral of 1/(sqrt(\sqrt{x)-1)}

\int \frac{1}{x - 1} d x

integral of-3(x+2)^{-2}

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

integral of ((x^2-3x+5)/(sqrt(x)))

\int (\frac{x ^{2} - 3 x + 5}{x}) d x

integral of (x+2)^2*ln(5x)

\int (x + 2)^{2} \cdot ln (5 x) d x

integral of tan(t)cot(t)

\int tan (t) cot (t) d t