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Beliebt Rechnen >

integral of (x-8)/(x^3-4x^2+4x)

Lösung

\int \frac{x - 8}{x ^{3} - 4 x ^{2} + 4 x} d x

Lösung

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

Schritte zur Lösung

\int \frac{x - 8}{x ^{3} - 4 x ^{2} + 4 x} d x

Ermittle den Partialbruch von

\frac{x - 8}{x ^{3} - 4 x ^{2} + 4 x} : - \frac{2}{x} + \frac{2}{x - 2} - \frac{3}{( x - 2 ) ^{2}}

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of tan^5(x)sec^5(x)

\int tan^{5} (x) sec^{5} (x) d x

integral of (cos(8sqrt(x)))/(sqrt(x))

\int \frac{cos ( 8 x )}{x} d x

integral of 12cos(4θ)

\int 12 cos (4 θ) d θ

integral of e^{6x}sin(e^{6x})

\int e^{6 x} sin (e^{6 x}) d x

integral of 1/(x^3sqrt(x))

\int \frac{1}{x ^{3} x} d x