de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (-3x^2)/(x^4-1)

Lösung

\int \frac{- 3 x ^{2}}{x ^{4} - 1} d x

Lösung

- 3 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣) + C

Schritte zur Lösung

\int \frac{- 3 x ^{2}}{x ^{4} - 1} d x

Entferne die Konstante:

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

= - 3 \cdot \int \frac{x ^{2}}{x ^{4} - 1} d x

Ermittle den Partialbruch von

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

= - 3 \cdot \int \frac{1}{2 ( x ^{2} + 1 )} - \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( 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

= - 3 (\int \frac{1}{2 ( x ^{2} + 1 )} d x - \int \frac{1}{4 ( x + 1 )} d x + \int \frac{1}{4 ( x - 1 )} d x)

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

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

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

= - 3 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣)

Füge eine Konstante zur Lösung hinzu

= - 3 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣) + C

Graph

Beliebte Beispiele

inverselaplace 1/(2s^2+3s+1)

in v erse l a pl a ce \frac{1}{2 s ^{2} + 3 s + 1}

derivative x^6cos(x)

d er i v a t i v e x^{6} cos (x)

derivative y(s)= 1/(s+ke^s)

d er i v a t i v e y (s) = \frac{1}{s + k e ^{s}}

sum from n=1 to infinity of arcsin(n)

n = 1 \sum \infty arcsin (n)

y^{''}+4y^'-5y=e^{3x}

y^{^{''}} + 4 y^{^{'}} - 5 y = e^{3 x}