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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{1}{( v - 1 ) v ^{2}} : \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}}

= \int \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}} d v

Wende die Summenregel an:

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

= \int \frac{1}{v - 1} d v - \int \frac{1}{v} d v - \int \frac{1}{v ^{2}} d v

\int \frac{1}{v - 1} d v = ln ∣ v - 1 ∣

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

= ln ∣ v - 1 ∣ - ln ∣ v ∣ - (- \frac{1}{v})

Ersetze zurück

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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