integral of (2x^3-7x^2+3x-2)/((x^2-x)^2)

Lösung

\int \frac{2 x ^{3} - 7 x ^{2} + 3 x - 2}{( x ^{2} - x ) ^{2}} d x

3 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - ln ∣ x ∣ + \frac{2}{x} + C

Schritte zur Lösung

\int \frac{2 x ^{3} - 7 x ^{2} + 3 x - 2}{( x ^{2} - x ) ^{2}} d x

Multipliziere aus

\frac{2 x ^{3} - 7 x ^{2} + 3 x - 2}{( x ^{2} - x ) ^{2}} : \frac{2 x ^{3}}{( x ^{2} - x ) ^{2}} - \frac{7 x ^{2}}{( x ^{2} - x ) ^{2}} + \frac{3 x}{( x ^{2} - x ) ^{2}} - \frac{2}{( x ^{2} - 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 ^{3}}{( x ^{2} - x ) ^{2}} d x - \int \frac{7 x ^{2}}{( x ^{2} - x ) ^{2}} d x + \int \frac{3 x}{( x ^{2} - x ) ^{2}} d x - \int \frac{2}{( x ^{2} - x ) ^{2}} d x

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

\int \frac{7 x ^{2}}{( x ^{2} - x ) ^{2}} d x = - \frac{7}{x - 1}

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

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

= 2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - (- \frac{7}{x - 1}) + 3 (ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 2 (2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1})

Vereinfache

2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - (- \frac{7}{x - 1}) + 3 (ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 2 (2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1}) : 3 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - ln ∣ x ∣ + \frac{2}{x}

= 3 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - ln ∣ x ∣ + \frac{2}{x}

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - ln ∣ x ∣ + \frac{2}{x} + C

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