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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{1}{u ^{3} ( u - 3 )} : - \frac{1}{27 u} - \frac{1}{9 u ^{2}} - \frac{1}{3 u ^{3}} + \frac{1}{27 ( u - 3 )}

= \int - \frac{1}{27 u} - \frac{1}{9 u ^{2}} - \frac{1}{3 u ^{3}} + \frac{1}{27 ( 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{1}{27 u} d u - \int \frac{1}{9 u ^{2}} d u - \int \frac{1}{3 u ^{3}} d u + \int \frac{1}{27 ( u - 3 )} d u

\int \frac{1}{27 u} d u = \frac{1}{27} ln ∣ u ∣

\int \frac{1}{9 u ^{2}} d u = - \frac{1}{9 u}

\int \frac{1}{3 u ^{3}} d u = - \frac{1}{6 u ^{2}}

\int \frac{1}{27 ( u - 3 )} d u = \frac{1}{27} ln ∣ u - 3 ∣

= - \frac{1}{27} ln ∣ u ∣ - (- \frac{1}{9 u}) - (- \frac{1}{6 u ^{2}}) + \frac{1}{27} ln ∣ u - 3 ∣

Setze in

u = x + 1

ein

= - \frac{1}{27} ln ∣ x + 1 ∣ - (- \frac{1}{9 ( x + 1 )}) - (- \frac{1}{6 ( x + 1 ) ^{2}}) + \frac{1}{27} ln ∣ x + 1 - 3 ∣

Vereinfache

- \frac{1}{27} ln ∣ x + 1 ∣ - (- \frac{1}{9 ( x + 1 )}) - (- \frac{1}{6 ( x + 1 ) ^{2}}) + \frac{1}{27} ln ∣ x + 1 - 3 ∣ : - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣

= - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣ + C

Graph

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