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

Lösung

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

Lösung

\frac{x ^{2}}{2} + 2 x + \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{4}}{( x ^{2} - 1 ) ( x - 2 )} : x + 2 + \frac{1}{6 ( x + 1 )} - \frac{1}{2 ( x - 1 )} + \frac{16}{3 ( x - 2 )}

= \int x + 2 + \frac{1}{6 ( x + 1 )} - \frac{1}{2 ( x - 1 )} + \frac{16}{3 ( x - 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 x d x + \int 2 d x + \int \frac{1}{6 ( x + 1 )} d x - \int \frac{1}{2 ( x - 1 )} d x + \int \frac{16}{3 ( x - 2 )} d x

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

\int 2 d x = 2 x

\int \frac{1}{6 ( x + 1 )} d x = \frac{1}{6} ln ∣ x + 1 ∣

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

\int \frac{16}{3 ( x - 2 )} d x = \frac{16}{3} ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 2 x + \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + 2 x + \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of (sinh(e^{-x/2}))/(e^{x/2)}

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

integral of rln((2r)/g)

\int r ln (\frac{2 r}{g}) d r

integral of (x^4-2x)^5(2x^3-1)

\int (x^{4} - 2 x)^{5} (2 x^{3} - 1) d x

integral of (cos(x)+x)

\int (cos (x) + x) d x

integral of (1+x^2)/(x^4)

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