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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{12}}{12} + x^{9} + 5 x^{6} + \frac{40 x ^{3}}{3} + \frac{16}{x ^{3}} + \frac{16}{3 x ^{6}} + C

Schritte zur Lösung

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

Multipliziere aus

(x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{5} : x^{11} + 9 x^{8} + 30 x^{5} + 40 x^{2} - \frac{48}{x ^{4}} - \frac{32}{x ^{7}}

= \int x^{11} + 9 x^{8} + 30 x^{5} + 40 x^{2} - \frac{48}{x ^{4}} - \frac{32}{x ^{7}} 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^{11} d x + \int 9 x^{8} d x + \int 30 x^{5} d x + \int 40 x^{2} d x - \int \frac{48}{x ^{4}} d x - \int \frac{32}{x ^{7}} d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 9 x^{8} d x = x^{9}

\int 30 x^{5} d x = 5 x^{6}

\int 40 x^{2} d x = \frac{40 x ^{3}}{3}

\int \frac{48}{x ^{4}} d x = - \frac{16}{x ^{3}}

\int \frac{32}{x ^{7}} d x = - \frac{16}{3 x ^{6}}

= \frac{x ^{12}}{12} + x^{9} + 5 x^{6} + \frac{40 x ^{3}}{3} - (- \frac{16}{x ^{3}}) - (- \frac{16}{3 x ^{6}})

Vereinfache

= \frac{x ^{12}}{12} + x^{9} + 5 x^{6} + \frac{40 x ^{3}}{3} + \frac{16}{x ^{3}} + \frac{16}{3 x ^{6}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{12}}{12} + x^{9} + 5 x^{6} + \frac{40 x ^{3}}{3} + \frac{16}{x ^{3}} + \frac{16}{3 x ^{6}} + C

Graph

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