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

Lösung

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

\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{x - 2} + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

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

Vereinfache

2 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{2 ( x - 2 )}) : \frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{x - 2}

= \frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{x - 2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{x - 2} + C

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