de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (2x^3-4x-8)/(x(x-1)(x^2+4))

Lösung

\int \frac{2 x ^{3} - 4 x - 8}{x ( x - 1 ) ( x ^{2} + 4 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{2 x ^{3} - 4 x - 8}{x ( x - 1 ) ( x ^{2} + 4 )} d x

Wende U-Substitution an

= \int \frac{- 4 u + 2 ( u + 1 ) ^{3} - 12}{u ( u + 1 ) ( u ^{2} + 2 u + 5 )} d u

Ermittle den Partialbruch von

\frac{- 4 u + 2 ( u + 1 ) ^{3} - 12}{u ( u + 1 ) ( u ^{2} + 2 u + 5 )} : - \frac{2}{u} + \frac{2}{u + 1} + \frac{2 u + 6}{u ^{2} + 2 u + 5}

= \int - \frac{2}{u} + \frac{2}{u + 1} + \frac{2 u + 6}{u ^{2} + 2 u + 5} d u

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{2}{u} d u + \int \frac{2}{u + 1} d u + \int \frac{2 u + 6}{u ^{2} + 2 u + 5} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{2}{u + 1} d u = 2 ln ∣ u + 1 ∣

\int \frac{2 u + 6}{u ^{2} + 2 u + 5} d u = ln u^{2} + 2 u + 5 + 2 arctan (\frac{1}{2} (u + 1))

= - 2 ln ∣ u ∣ + 2 ln ∣ u + 1 ∣ + ln u^{2} + 2 u + 5 + 2 arctan (\frac{1}{2} (u + 1))

Setze in

u = x - 1

ein

= - 2 ln ∣ x - 1 ∣ + 2 ln ∣ x - 1 + 1 ∣ + ln (x - 1)^{2} + 2 (x - 1) + 5 + 2 arctan (\frac{1}{2} (x - 1 + 1))

Vereinfache

- 2 ln ∣ x - 1 ∣ + 2 ln ∣ x - 1 + 1 ∣ + ln (x - 1)^{2} + 2 (x - 1) + 5 + 2 arctan (\frac{1}{2} (x - 1 + 1)) : - 2 ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + ln x^{2} + 4 + 2 arctan (\frac{x}{2})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (8x)/(1+4x^2)

\int \frac{8 x}{1 + 4 x ^{2}} d x

integral of 5/(x(x^2+4)^2)

\int \frac{5}{x ( x ^{2} + 4 ) ^{2}} d x

integral of 7x^6(x^7+1)^8

\int 7 x^{6} (x^{7} + 1)^{8} d x

integral of x^2sqrt(25+x^2)

\int x^{2} 25 + x^{2} d x

integral of x^{-4}(x-2)^2

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