integral of 1/(x(1+x^8))

Lösung

\int \frac{1}{x ( 1 + x ^{8} )} d x

\frac{1}{8} (8 ln (x) - ln 1 + x^{8}) + C

Schritte zur Lösung

\int \frac{1}{x ( 1 + x ^{8} )} d x

Wende U-Substitution an

= \int \frac{1}{8 ( u - 1 ) u} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int \frac{1}{( u - 1 ) u} d u

Ermittle den Partialbruch von

\frac{1}{( u - 1 ) u} : \frac{1}{u - 1} - \frac{1}{u}

= \frac{1}{8} \cdot \int \frac{1}{u - 1} - \frac{1}{u} 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}{8} (\int \frac{1}{u - 1} d u - \int \frac{1}{u} d u)

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{8} (ln ∣ u - 1 ∣ - ln ∣ u ∣)

Setze in

u = 1 + x^{8}

ein

= \frac{1}{8} (ln 1 + x^{8} - 1 - ln 1 + x^{8})

Vereinfache

\frac{1}{8} (ln 1 + x^{8} - 1 - ln 1 + x^{8}) : \frac{1}{8} (8 ln (x) - ln 1 + x^{8})

= \frac{1}{8} (8 ln (x) - ln 1 + x^{8})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (8 ln (x) - ln 1 + x^{8}) + C

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