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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{4 u}{( u - 1 ) ^{3} - 6 ( u - 1 ) ^{2} + 8 u - 8} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{u}{( u - 1 ) ^{3} - 6 ( u - 1 ) ^{2} + 8 u - 8} d u

Ermittle den Partialbruch von

\frac{u}{( u - 1 ) ^{3} - 6 ( u - 1 ) ^{2} + 8 u - 8} : \frac{1}{8 ( u - 1 )} - \frac{3}{4 ( u - 3 )} + \frac{5}{8 ( u - 5 )}

= 4 \cdot \int \frac{1}{8 ( u - 1 )} - \frac{3}{4 ( u - 3 )} + \frac{5}{8 ( 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

= 4 (\int \frac{1}{8 ( u - 1 )} d u - \int \frac{3}{4 ( u - 3 )} d u + \int \frac{5}{8 ( u - 5 )} d u)

\int \frac{1}{8 ( u - 1 )} d u = \frac{1}{8} ln ∣ u - 1 ∣

\int \frac{3}{4 ( u - 3 )} d u = \frac{3}{4} ln ∣ u - 3 ∣

\int \frac{5}{8 ( u - 5 )} d u = \frac{5}{8} ln ∣ u - 5 ∣

= 4 (\frac{1}{8} ln ∣ u - 1 ∣ - \frac{3}{4} ln ∣ u - 3 ∣ + \frac{5}{8} ln ∣ u - 5 ∣)

Setze in

u = 2 x + 1

ein

= 4 (\frac{1}{8} ln ∣ 2 x + 1 - 1 ∣ - \frac{3}{4} ln ∣ 2 x + 1 - 3 ∣ + \frac{5}{8} ln ∣ 2 x + 1 - 5 ∣)

Vereinfache

4 (\frac{1}{8} ln ∣ 2 x + 1 - 1 ∣ - \frac{3}{4} ln ∣ 2 x + 1 - 3 ∣ + \frac{5}{8} ln ∣ 2 x + 1 - 5 ∣) : \frac{1}{2} ln (2 ∣ x ∣) - 3 ln ∣ 2 x - 2 ∣ + \frac{5}{2} ln ∣ 2 x - 4 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(4-5sin(x))

\int \frac{1}{4 - 5 sin ( x )} d x

integral of (5x+3)/((x+2)^2(x-1)^2)

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

integral of sec^2(2x)tan(2x)

\int sec^{2} (2 x) tan (2 x) d x

integral of 1/(1+sqrt(2)x)

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

integral of ((7x-1))/((x-1)(x-2)(x-3))

\int \frac{( 7 x - 1 )}{( x - 1 ) ( x - 2 ) ( x - 3 )} d x