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Beliebt Rechnen >

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

Lösung

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

Lösung

3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{5 u - 9}{u ( u - 3 )} d u

Ermittle den Partialbruch von

\frac{5 u - 9}{u ( u - 3 )} : \frac{3}{u} + \frac{2}{u - 3}

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

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

= 3 ln ∣ u ∣ + 2 ln ∣ u - 3 ∣

Setze in

u = x + 2

ein

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x + 2 - 3 ∣

Vereinfache

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

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

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

integral of 3/7 x^{7/4}

\int \frac{3}{7} x^{\frac{7}{4}} d x

integral of (5-4x)^4

\int (5 - 4 x)^{4} d x

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\int 12 - 4 x - x d x

integral of 300+20x-12x^2

\int 300 + 20 x - 12 x^{2} d x