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Beliebt Rechnen >

integral of 6/(x(x-6)^2)

Lösung

\int \frac{6}{x ( x - 6 ) ^{2}} d x

Lösung

\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6} + C

Schritte zur Lösung

\int \frac{6}{x ( x - 6 ) ^{2}} d x

Entferne die Konstante:

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

= 6 \cdot \int \frac{1}{x ( x - 6 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{x ( x - 6 ) ^{2}} : \frac{1}{36 x} - \frac{1}{36 ( x - 6 )} + \frac{1}{6 ( x - 6 ) ^{2}}

= 6 \cdot \int \frac{1}{36 x} - \frac{1}{36 ( x - 6 )} + \frac{1}{6 ( x - 6 ) ^{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

= 6 (\int \frac{1}{36 x} d x - \int \frac{1}{36 ( x - 6 )} d x + \int \frac{1}{6 ( x - 6 ) ^{2}} d x)

\int \frac{1}{36 x} d x = \frac{1}{36} ln ∣ x ∣

\int \frac{1}{36 ( x - 6 )} d x = \frac{1}{36} ln ∣ x - 6 ∣

\int \frac{1}{6 ( x - 6 ) ^{2}} d x = - \frac{1}{6 ( x - 6 )}

= 6 (\frac{1}{36} ln ∣ x ∣ - \frac{1}{36} ln ∣ x - 6 ∣ - \frac{1}{6 ( x - 6 )})

Vereinfache

6 (\frac{1}{36} ln ∣ x ∣ - \frac{1}{36} ln ∣ x - 6 ∣ - \frac{1}{6 ( x - 6 )}) : \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6}

= \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6} + C

Graph

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