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Beliebt Rechnen >

integral of 1/(x(x^{10)+2)}

Lösung

\int \frac{1}{x ( x ^{10} + 2 )} d x

Lösung

\frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2) + C

Schritte zur Lösung

\int \frac{1}{x ( x ^{10} + 2 )} d x

Wende U-Substitution an

= \int \frac{1}{10 u ( u + 2 )} d u

Entferne die Konstante:

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

= \frac{1}{10} \cdot \int \frac{1}{u ( u + 2 )} d u

Ermittle den Partialbruch von

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

= \frac{1}{10} \cdot \int \frac{1}{2 u} - \frac{1}{2 ( u + 2 )} 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}{10} (\int \frac{1}{2 u} d u - \int \frac{1}{2 ( u + 2 )} d u)

\int \frac{1}{2 u} d u = \frac{1}{2} ln ∣ u ∣

\int \frac{1}{2 ( u + 2 )} d u = \frac{1}{2} ln ∣ u + 2 ∣

= \frac{1}{10} (\frac{1}{2} ln ∣ u ∣ - \frac{1}{2} ln ∣ u + 2 ∣)

Setze in

u = x^{10}

ein

= \frac{1}{10} (\frac{1}{2} ln x^{10} - \frac{1}{2} ln x^{10} + 2)

\frac{1}{2} ln x^{10} = 5 ln (x)

= \frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2) + C

Graph

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