integral of 1/(x(x^2-2)^2)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Setze in

u = x^{2}

ein

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

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

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

Füge eine Konstante zur Lösung hinzu

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