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Beliebt Rechnen >

integral of (x^4-6x^3+12x^2+6)/(x^3-6x^2+12x-8)

Lösung

\int \frac{x ^{4} - 6 x ^{3} + 12 x ^{2} + 6}{x ^{3} - 6 x ^{2} + 12 x - 8} d x

Lösung

\frac{x ^{2}}{2} - \frac{8}{x - 2} - \frac{11}{( x - 2 ) ^{2}} + C

Schritte zur Lösung

\int \frac{x ^{4} - 6 x ^{3} + 12 x ^{2} + 6}{x ^{3} - 6 x ^{2} + 12 x - 8} d x

Ermittle den Partialbruch von

\frac{x ^{4} - 6 x ^{3} + 12 x ^{2} + 6}{x ^{3} - 6 x ^{2} + 12 x - 8} : x + \frac{8}{( x - 2 ) ^{2}} + \frac{22}{( x - 2 ) ^{3}}

= \int x + \frac{8}{( x - 2 ) ^{2}} + \frac{22}{( x - 2 ) ^{3}} 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{8}{( x - 2 ) ^{2}} d x + \int \frac{22}{( x - 2 ) ^{3}} d x

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

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

\int \frac{22}{( x - 2 ) ^{3}} d x = - \frac{11}{( x - 2 ) ^{2}}

= \frac{x ^{2}}{2} - \frac{8}{x - 2} - \frac{11}{( x - 2 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{8}{x - 2} - \frac{11}{( x - 2 ) ^{2}} + C

Graph

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