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Beliebt Rechnen >

integral of (x^2)/((x-1)^{20)}

Lösung

\int \frac{x ^{2}}{( x - 1 ) ^{20}} d x

Lösung

- \frac{1}{17 ( x - 1 ) ^{17}} - \frac{1}{9 ( x - 1 ) ^{18}} - \frac{1}{19 ( x - 1 ) ^{19}} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( x - 1 ) ^{20}} d x

Wende U-Substitution an

= \int \frac{( u + 1 ) ^{2}}{u ^{20}} d u

Multipliziere aus

\frac{( u + 1 ) ^{2}}{u ^{20}} : \frac{1}{u ^{18}} + \frac{2}{u ^{19}} + \frac{1}{u ^{20}}

= \int \frac{1}{u ^{18}} + \frac{2}{u ^{19}} + \frac{1}{u ^{20}} 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}{u ^{18}} d u + \int \frac{2}{u ^{19}} d u + \int \frac{1}{u ^{20}} d u

\int \frac{1}{u ^{18}} d u = - \frac{1}{17 u ^{17}}

\int \frac{2}{u ^{19}} d u = - \frac{1}{9 u ^{18}}

\int \frac{1}{u ^{20}} d u = - \frac{1}{19 u ^{19}}

= - \frac{1}{17 u ^{17}} - \frac{1}{9 u ^{18}} - \frac{1}{19 u ^{19}}

Setze in

u = x - 1

ein

= - \frac{1}{17 ( x - 1 ) ^{17}} - \frac{1}{9 ( x - 1 ) ^{18}} - \frac{1}{19 ( x - 1 ) ^{19}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{17 ( x - 1 ) ^{17}} - \frac{1}{9 ( x - 1 ) ^{18}} - \frac{1}{19 ( x - 1 ) ^{19}} + C

Graph

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