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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2}}{- x ^{2} + 4 x - 4} : - 1 - \frac{4}{x - 2} - \frac{4}{( x - 2 ) ^{2}}

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

\int 1 d x = x

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

\int \frac{4}{( x - 2 ) ^{2}} d x = - \frac{4}{x - 2}

= - x - 4 ln ∣ x - 2 ∣ - (- \frac{4}{x - 2})

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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