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Beliebt Rechnen >

integral of (4x+10)/((x-2)(x-1)^2)

Lösung

\int \frac{4 x + 10}{( x - 2 ) ( x - 1 ) ^{2}} d x

Lösung

\frac{14}{x - 1} + 18 ln ∣ x - 2 ∣ - 18 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{4 x + 10}{( x - 2 ) ( x - 1 ) ^{2}} d x

Multipliziere aus

\frac{4 x + 10}{( x - 2 ) ( x - 1 ) ^{2}} : \frac{4 x}{( x - 2 ) ( x - 1 ) ^{2}} + \frac{10}{( x - 2 ) ( 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{4 x}{( x - 2 ) ( x - 1 ) ^{2}} d x + \int \frac{10}{( x - 2 ) ( x - 1 ) ^{2}} d x

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

\int \frac{10}{( x - 2 ) ( x - 1 ) ^{2}} d x = 10 (ln ∣ x - 2 ∣ - ln ∣ x - 1 ∣ + \frac{1}{x - 1})

= 4 (\frac{1}{x - 1} + 2 ln ∣ x - 2 ∣ - 2 ln ∣ x - 1 ∣) + 10 (ln ∣ x - 2 ∣ - ln ∣ x - 1 ∣ + \frac{1}{x - 1})

Vereinfache

4 (\frac{1}{x - 1} + 2 ln ∣ x - 2 ∣ - 2 ln ∣ x - 1 ∣) + 10 (ln ∣ x - 2 ∣ - ln ∣ x - 1 ∣ + \frac{1}{x - 1}) : \frac{14}{x - 1} + 18 ln ∣ x - 2 ∣ - 18 ln ∣ x - 1 ∣

= \frac{14}{x - 1} + 18 ln ∣ x - 2 ∣ - 18 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{14}{x - 1} + 18 ln ∣ x - 2 ∣ - 18 ln ∣ x - 1 ∣ + C

Graph

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