integral of (8x^3-8x+5)/(x^3-x^2)

Lösung

\int \frac{8 x ^{3} - 8 x + 5}{x ^{3} - x ^{2}} d x

\frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x} + C

Schritte zur Lösung

\int \frac{8 x ^{3} - 8 x + 5}{x ^{3} - x ^{2}} d x

Wende die partielle Integration an

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} - \int - \frac{8 x ^{3} - 16 x ^{2} + 23 x - 10}{x ^{2} ( x - 1 ) ^{2}} d x

\int - \frac{8 x ^{3} - 16 x ^{2} + 23 x - 10}{x ^{2} ( x - 1 ) ^{2}} d x = - 5 ln ∣ x - 1 ∣ + \frac{5}{x - 1} - 3 ln ∣ x ∣ - \frac{10}{x}

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} - (- 5 ln ∣ x - 1 ∣ + \frac{5}{x - 1} - 3 ln ∣ x ∣ - \frac{10}{x})

Vereinfache

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x} + C

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