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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{1}{u ( u - 3 ) ^{2}} : \frac{1}{9 u} - \frac{1}{9 ( u - 3 )} + \frac{1}{3 ( u - 3 ) ^{2}}

= \int \frac{1}{9 u} - \frac{1}{9 ( u - 3 )} + \frac{1}{3 ( u - 3 ) ^{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{1}{9 u} d u - \int \frac{1}{9 ( u - 3 )} d u + \int \frac{1}{3 ( u - 3 ) ^{2}} d u

\int \frac{1}{9 u} d u = \frac{1}{9} ln ∣ u ∣

\int \frac{1}{9 ( u - 3 )} d u = \frac{1}{9} ln ∣ u - 3 ∣

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

= \frac{1}{9} ln ∣ u ∣ - \frac{1}{9} ln ∣ u - 3 ∣ - \frac{1}{3 ( u - 3 )}

Setze in

u = x + 2

ein

= \frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x + 2 - 3 ∣ - \frac{1}{3 ( x + 2 - 3 )}

Vereinfache

\frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x + 2 - 3 ∣ - \frac{1}{3 ( x + 2 - 3 )} : \frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )}

= \frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} ln ∣ x + 2 ∣ - \frac{1}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )} + C

Graph

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