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Beliebt Rechnen >

integral of (x^3-6x^2+4x-10)/(x^2-1)

Lösung

\int \frac{x ^{3} - 6 x ^{2} + 4 x - 10}{x ^{2} - 1} d x

Lösung

\frac{1}{2} x^{2} - 6 x + \frac{5}{2} ln x^{2} - 1 + 8 ln ∣ x + 1 ∣ - 8 ln ∣ x - 1 ∣ - \frac{1}{2} + C

Schritte zur Lösung

\int \frac{x ^{3} - 6 x ^{2} + 4 x - 10}{x ^{2} - 1} d x

Multipliziere aus

\frac{x ^{3} - 6 x ^{2} + 4 x - 10}{x ^{2} - 1} : \frac{x ^{3}}{x ^{2} - 1} - \frac{6 x ^{2}}{x ^{2} - 1} + \frac{4 x}{x ^{2} - 1} - \frac{10}{x ^{2} - 1}

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 ^{3}}{x ^{2} - 1} d x - \int \frac{6 x ^{2}}{x ^{2} - 1} d x + \int \frac{4 x}{x ^{2} - 1} d x - \int \frac{10}{x ^{2} - 1} d x

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

\int \frac{6 x ^{2}}{x ^{2} - 1} d x = - 6 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - x)

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

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

= \frac{1}{2} (x^{2} - 1 + ln x^{2} - 1) - (- 6 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - x)) + 2 ln x^{2} - 1 - (- 10 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣))

Vereinfache

\frac{1}{2} (x^{2} - 1 + ln x^{2} - 1) - (- 6 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - x)) + 2 ln x^{2} - 1 - (- 10 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)) : \frac{1}{2} x^{2} - 6 x + \frac{5}{2} ln x^{2} - 1 + 8 ln ∣ x + 1 ∣ - 8 ln ∣ x - 1 ∣ - \frac{1}{2}

= \frac{1}{2} x^{2} - 6 x + \frac{5}{2} ln x^{2} - 1 + 8 ln ∣ x + 1 ∣ - 8 ln ∣ x - 1 ∣ - \frac{1}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} - 6 x + \frac{5}{2} ln x^{2} - 1 + 8 ln ∣ x + 1 ∣ - 8 ln ∣ x - 1 ∣ - \frac{1}{2} + C

Graph

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