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Beliebt Rechnen >

integral of (x^4)/((x-1)^2(x+2))

Lösung

\int \frac{x ^{4}}{( x - 1 ) ^{2} ( x + 2 )} d x

Lösung

\frac{x ^{2}}{2} + \frac{11}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )} + \frac{16}{9} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

\int \frac{x ^{4}}{( x - 1 ) ^{2} ( x + 2 )} d x

Ermittle den Partialbruch von

\frac{x ^{4}}{( x - 1 ) ^{2} ( x + 2 )} : x + \frac{11}{9 ( x - 1 )} + \frac{1}{3 ( x - 1 ) ^{2}} + \frac{16}{9 ( x + 2 )}

= \int x + \frac{11}{9 ( x - 1 )} + \frac{1}{3 ( x - 1 ) ^{2}} + \frac{16}{9 ( 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 x d x + \int \frac{11}{9 ( x - 1 )} d x + \int \frac{1}{3 ( x - 1 ) ^{2}} d x + \int \frac{16}{9 ( x + 2 )} d x

\int x d x = \frac{x ^{2}}{2}

\int \frac{11}{9 ( x - 1 )} d x = \frac{11}{9} ln ∣ x - 1 ∣

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

\int \frac{16}{9 ( x + 2 )} d x = \frac{16}{9} ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} + \frac{11}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )} + \frac{16}{9} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + \frac{11}{9} ln ∣ x - 1 ∣ - \frac{1}{3 ( x - 1 )} + \frac{16}{9} ln ∣ x + 2 ∣ + C

Graph

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