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Beliebt Rechnen >

integral of (x^2)/((1-x)^{100)}

Lösung

\int \frac{x ^{2}}{( 1 - x ) ^{100}} d x

Lösung

\frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( 1 - x ) ^{100}} d x

Wende U-Substitution an

= \int - \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

Multipliziere aus

\frac{( - u + 1 ) ^{2}}{u ^{100}} : \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}}

= - \int \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}} 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 ^{98}} d u - \int \frac{2}{u ^{99}} d u + \int \frac{1}{u ^{100}} d u)

\int \frac{1}{u ^{98}} d u = - \frac{1}{97 u ^{97}}

\int \frac{2}{u ^{99}} d u = - \frac{1}{49 u ^{98}}

\int \frac{1}{u ^{100}} d u = - \frac{1}{99 u ^{99}}

= - (- \frac{1}{97 u ^{97}} - (- \frac{1}{49 u ^{98}}) - \frac{1}{99 u ^{99}})

Setze in

u = 1 - x

ein

= - (- \frac{1}{97 ( 1 - x ) ^{97}} - (- \frac{1}{49 ( 1 - x ) ^{98}}) - \frac{1}{99 ( 1 - x ) ^{99}})

Vereinfache

= \frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}} + C

Graph

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