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Beliebt Rechnen >

integral of ((2x-3))/((x-1)^2(x-2)^2)

Lösung

\int \frac{( 2 x - 3 )}{( x - 1 ) ^{2} ( x - 2 ) ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{2 x - 3}{( x - 1 ) ^{2} ( x - 2 ) ^{2}} d x

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

Setze in

u = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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