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Beliebt Rechnen >

integral of (x^2-x+2)/(3x^3-2x^2+6x-4)

Lösung

\int \frac{x ^{2} - x + 2}{3 x ^{3} - 2 x ^{2} + 6 x - 4} d x

Lösung

\frac{8}{33} ln ∣ 3 x - 2 ∣ + \frac{1}{11} (\frac{1}{2} ln x^{2} + 2 - \frac{3}{2} arctan (\frac{x}{2})) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 2}{3 x ^{3} - 2 x ^{2} + 6 x - 4} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 2}{3 x ^{3} - 2 x ^{2} + 6 x - 4} : \frac{8}{11 ( 3 x - 2 )} + \frac{x - 3}{11 ( x ^{2} + 2 )}

= \int \frac{8}{11 ( 3 x - 2 )} + \frac{x - 3}{11 ( x ^{2} + 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{8}{11 ( 3 x - 2 )} d x + \int \frac{x - 3}{11 ( x ^{2} + 2 )} d x

\int \frac{8}{11 ( 3 x - 2 )} d x = \frac{8}{33} ln ∣ 3 x - 2 ∣

\int \frac{x - 3}{11 ( x ^{2} + 2 )} d x = \frac{1}{11} (\frac{1}{2} ln x^{2} + 2 - \frac{3}{2} arctan (\frac{x}{2}))

= \frac{8}{33} ln ∣ 3 x - 2 ∣ + \frac{1}{11} (\frac{1}{2} ln x^{2} + 2 - \frac{3}{2} arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{8}{33} ln ∣ 3 x - 2 ∣ + \frac{1}{11} (\frac{1}{2} ln x^{2} + 2 - \frac{3}{2} arctan (\frac{x}{2})) + C

Graph

Beliebte Beispiele

fläche y=(x^2)/2+2,[-3,0]

a re a y = \frac{x ^{2}}{2} + 2, [- 3, 0]

derivative of (3x+6^2)

\frac{d}{d x} ((3 x + 6)^{2})

fläche y=-x^2-2x+3,y=-x+1

a re a y = - x^{2} - 2 x + 3, y = - x + 1

integral of ((x-1))/((x^2-2x-35)^7)

\int \frac{( x - 1 )}{( x ^{2} - 2 x - 35 ) ^{7}} d x

derivative 9/(sqrt(t))

d er i v a t i v e \frac{9}{t}