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Beliebt Rechnen >

integral of x/(x^8-1)

Lösung

\int \frac{x}{x ^{8} - 1} d x

Lösung

\frac{1}{2} (- \frac{1}{2} arctan (x^{2}) - \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln x^{2} - 1) + C

Schritte zur Lösung

\int \frac{x}{x ^{8} - 1} d x

Wende U-Substitution an

= \int \frac{1}{2 ( u ^{4} - 1 )} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{4} - 1} d u

Ermittle den Partialbruch von

\frac{1}{u ^{4} - 1} : - \frac{1}{2 ( u ^{2} + 1 )} - \frac{1}{4 ( u + 1 )} + \frac{1}{4 ( u - 1 )}

= \frac{1}{2} \cdot \int - \frac{1}{2 ( u ^{2} + 1 )} - \frac{1}{4 ( u + 1 )} + \frac{1}{4 ( u - 1 )} 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}{2} (- \int \frac{1}{2 ( u ^{2} + 1 )} d u - \int \frac{1}{4 ( u + 1 )} d u + \int \frac{1}{4 ( u - 1 )} d u)

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

\int \frac{1}{4 ( u + 1 )} d u = \frac{1}{4} ln ∣ u + 1 ∣

\int \frac{1}{4 ( u - 1 )} d u = \frac{1}{4} ln ∣ u - 1 ∣

= \frac{1}{2} (- \frac{1}{2} arctan (u) - \frac{1}{4} ln ∣ u + 1 ∣ + \frac{1}{4} ln ∣ u - 1 ∣)

Setze in

u = x^{2}

ein

= \frac{1}{2} (- \frac{1}{2} arctan (x^{2}) - \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln x^{2} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (- \frac{1}{2} arctan (x^{2}) - \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln x^{2} - 1) + C

Graph

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