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^3-2x^2+5x-2)/(x^3-2x^2+x)

Lösung

\int \frac{x ^{3} - 2 x ^{2} + 5 x - 2}{x ^{3} - 2 x ^{2} + x} d x

Lösung

\frac{x ^{3}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{10}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - 2 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{3} - 2 x ^{2} + 5 x - 2}{x ^{3} - 2 x ^{2} + x} d x

Multipliziere aus

\frac{x ^{3} - 2 x ^{2} + 5 x - 2}{x ^{3} - 2 x ^{2} + x} : \frac{x ^{3}}{x ^{3} - 2 x ^{2} + x} - \frac{2 x ^{2}}{x ^{3} - 2 x ^{2} + x} + \frac{5 x}{x ^{3} - 2 x ^{2} + x} - \frac{2}{x ^{3} - 2 x ^{2} + 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{x ^{3}}{x ^{3} - 2 x ^{2} + x} d x - \int \frac{2 x ^{2}}{x ^{3} - 2 x ^{2} + x} d x + \int \frac{5 x}{x ^{3} - 2 x ^{2} + x} d x - \int \frac{2}{x ^{3} - 2 x ^{2} + x} d x

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

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

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

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

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}} - 2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) + 5 (\frac{x}{x ^{2} - 2 x + 1} - \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}}) - 2 (ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1})

Vereinfache

\frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}} - 2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) + 5 (\frac{x}{x ^{2} - 2 x + 1} - \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}}) - 2 (ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1}) : \frac{x ^{3}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{10}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - 2 ln ∣ x ∣

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{10}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{10}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - 2 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

derivative of-2sin^2(x)

\frac{d}{d x} (- 2 sin^{2} (x))

sum from n=1 to infinity of ((ln(3n)))/n

n = 1 \sum \infty \frac{( ln ( 3 n ) )}{n}

derivative (2x-3)^6

d er i v a t i v e (2 x - 3)^{6}

erweitern \sqrt[3]{x}(2+x)

e x p an d 3 x (2 + x)

limit as x approaches 0+of e^{cot(x)}

x \to 0 + lim (e^{c o t (x)})