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Beliebt Rechnen >

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

Lösung

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

Lösung

6 ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{25}{3} ln ∣ 3 x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{- 7 x ^{2} + 2 x - 1}{( x - 1 ) ^{2} ( 3 x - 2 )} : \frac{6}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - \frac{25}{3 x - 2}

= \int \frac{6}{x - 1} - \frac{6}{( x - 1 ) ^{2}} - \frac{25}{3 x - 2} d 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{6}{x - 1} d x - \int \frac{6}{( x - 1 ) ^{2}} d x - \int \frac{25}{3 x - 2} d x

\int \frac{6}{x - 1} d x = 6 ln ∣ x - 1 ∣

\int \frac{6}{( x - 1 ) ^{2}} d x = - \frac{6}{x - 1}

\int \frac{25}{3 x - 2} d x = \frac{25}{3} ln ∣ 3 x - 2 ∣

= 6 ln ∣ x - 1 ∣ - (- \frac{6}{x - 1}) - \frac{25}{3} ln ∣ 3 x - 2 ∣

Vereinfache

= 6 ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{25}{3} ln ∣ 3 x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{25}{3} ln ∣ 3 x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of (a^xln(a))/(1+a^{2x)}

\int \frac{a ^{x} ln ( a )}{1 + a ^{2 x}} d x

integral of cos(5x-2)

\int cos (5 x - 2) d x

integral of f(ax+b)

\int f (a x + b) d x

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

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

integral of 1/pi

\int \frac{1}{π} d x