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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

\int \frac{1}{3 u + 2} d u = \frac{1}{3} ln ∣ 3 u + 2 ∣

= - 2 ln ∣ u ∣ - (- \frac{1}{u}) + \frac{1}{3} ln ∣ 3 u + 2 ∣

Setze in

u = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

fläche 7x+y^2=15,x=2y

a re a 7 x + y^{2} = 15, x = 2 y

fläche y=cos(3x),(0.4,0.6)

a re a y = cos (3 x), (0.4, 0.6)

limit as x approaches-3 of 2x+5

x \to - 3 lim (2 x + 5)

integral of (-6x^2)/(x^4-1)

\int \frac{- 6 x ^{2}}{x ^{4} - 1} d x

integral of xln^2(x)

\int x ln^{2} (x) d x