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Beliebt Rechnen >

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

Lösung

\int \frac{2 - x ^{2}}{x - x ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{2 - x ^{2}}{x - x ^{2}} d x

Multipliziere aus

\frac{2 - x ^{2}}{x - x ^{2}} : \frac{2}{x - x ^{2}} - \frac{x ^{2}}{x - x ^{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 ^{2}} d x - \int \frac{x ^{2}}{x - x ^{2}} d x

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 3x^5-6x^2+x-9

\int 3 x^{5} - 6 x^{2} + x - 9 d x

integral of 3x^2sqrt(x^3+4)

\int 3 x^{2} x^{3} + 4 d x

integral of (2x^2)/(a^2+x^2)

\int \frac{2 x ^{2}}{a ^{2} + x ^{2}} d x

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\int (x^{- 3} - 6 x^{- 2} + 10) d x

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

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