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Beliebt Rechnen >

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

Lösung

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

Lösung

- 3 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 4 ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 4 (ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1}) : - 3 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 4 ln ∣ x ∣

= - 3 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 4 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - 3 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 4 ln ∣ x ∣ + C

Graph

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