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+1)/(x^3+2x^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} + x + 1}{x ^{3} + 2 x ^{2}} : \frac{x ^{2}}{x ^{3} + 2 x ^{2}} + \frac{x}{x ^{3} + 2 x ^{2}} + \frac{1}{x ^{3} + 2 x ^{2}}

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

\int \frac{x ^{2}}{x ^{3} + 2 x ^{2}} d x = \frac{x}{x + 2} + ln ∣ x + 2 ∣ + \frac{2}{x + 2}

\int \frac{x}{x ^{3} + 2 x ^{2}} d x = \frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣

\int \frac{1}{x ^{3} + 2 x ^{2}} d x = - \frac{1}{4} ln ∣ x ∣ - \frac{1}{2 x} + \frac{1}{4} ln ∣ x + 2 ∣

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

Vereinfache

\frac{x}{x + 2} + ln ∣ x + 2 ∣ + \frac{2}{x + 2} + \frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{4} ln ∣ x ∣ - \frac{1}{2 x} + \frac{1}{4} ln ∣ x + 2 ∣ : \frac{1}{4} ln ∣ x ∣ + \frac{3}{4} ln ∣ x + 2 ∣ + \frac{x}{x + 2} + \frac{2}{x + 2} - \frac{1}{2 x}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

(\partial)/(\partial y)(xe^xcos(y))

\frac{\partial}{\partial y} (x e^{x} cos (y))

f(x)=2sin(2x)-ln(3x+1)

f (x) = 2 sin (2 x) - ln (3 x + 1)

integral of cos(x)-cos^2(x)

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

integral from 2 to 4 of 2pix(-3/2 x+6)

\int_{2}^{4} 2 π x (- \frac{3}{2} x + 6) d x

integral of x/2 e^{-x/2}

\int \frac{x}{2} e^{- \frac{x}{2}} d x