integral of (x^{11})/((x^6+1)^{100)}

Lösung

\int \frac{x ^{11}}{( x ^{6} + 1 ) ^{100}} d x

\frac{- 99 x ^{6} - 1}{58212 ( x ^{6} + 1 ) ^{99}} + C

Schritte zur Lösung

\int \frac{x ^{11}}{( x ^{6} + 1 ) ^{100}} d x

Wende U-Substitution an

= \int \frac{u - 1}{6 u ^{100}} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{u - 1}{u ^{100}} d u

Multipliziere aus

\frac{u - 1}{u ^{100}} : \frac{1}{u ^{99}} - \frac{1}{u ^{100}}

= \frac{1}{6} \cdot \int \frac{1}{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

= \frac{1}{6} (\int \frac{1}{u ^{99}} d u - \int \frac{1}{u ^{100}} d u)

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

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

= \frac{1}{6} (- \frac{1}{98 u ^{98}} - (- \frac{1}{99 u ^{99}}))

Setze in

u = x^{6} + 1

ein

= \frac{1}{6} (- \frac{1}{98 ( x ^{6} + 1 ) ^{98}} - (- \frac{1}{99 ( x ^{6} + 1 ) ^{99}}))

Vereinfache

\frac{1}{6} (- \frac{1}{98 ( x ^{6} + 1 ) ^{98}} - (- \frac{1}{99 ( x ^{6} + 1 ) ^{99}})) : \frac{- 99 x ^{6} - 1}{58212 ( x ^{6} + 1 ) ^{99}}

= \frac{- 99 x ^{6} - 1}{58212 ( x ^{6} + 1 ) ^{99}}

Füge eine Konstante zur Lösung hinzu

= \frac{- 99 x ^{6} - 1}{58212 ( x ^{6} + 1 ) ^{99}} + C

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