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

Lösung

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

2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

= 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1}

Füge eine Konstante zur Lösung hinzu

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

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