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)/(x^4-2x^2+1)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(e^{-x)-e^x}

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

integral of c/(1+y^2)

\int \frac{c}{1 + y ^{2}} d y

integral of 5/(xln(x^3))

\int \frac{5}{x ln ( x ^{3} )} d x

integral of 1/(sqrt(2(x-3)^2+1))

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

integral of xsqrt(x)+1/(x^3)

\int x x + \frac{1}{x ^{3}} d x