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Beliebt Rechnen >

integral of (-x+8)/(x^2-6x+8)

Lösung

\int \frac{- x + 8}{x ^{2} - 6 x + 8} d x

Lösung

- \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{- x + 8}{x ^{2} - 6 x + 8} d x

Multipliziere aus

\frac{- x + 8}{x ^{2} - 6 x + 8} : - \frac{x}{x ^{2} - 6 x + 8} + \frac{8}{x ^{2} - 6 x + 8}

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}{x ^{2} - 6 x + 8} d x + \int \frac{8}{x ^{2} - 6 x + 8} d x

\int \frac{x}{x ^{2} - 6 x + 8} d x = \frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)

\int \frac{8}{x ^{2} - 6 x + 8} d x = 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣)

= - (\frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)) + 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣)

Vereinfache

- (\frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)) + 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣) : - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣

= - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣ + C

Graph

Beliebte Beispiele

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

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\int \frac{1}{x 64 - x ^{2}} d x

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\int (3 y + 8)^{5} d y