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Beliebt Rechnen >

integral of (4x^2-8x)/((x-1)^2(x^2+1)^2)

Lösung

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

Lösung

2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} - ln x^{2} + 1 + arctan (x) + \frac{2 x}{x ^{2} + 1} + \frac{1}{x ^{2} + 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{4 x ^{2} - 8 x}{( x - 1 ) ^{2} ( x ^{2} + 1 ) ^{2}} : \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + \frac{- 2 x - 1}{x ^{2} + 1} + \frac{- 2 x + 4}{( x ^{2} + 1 ) ^{2}}

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} - ln x^{2} + 1 + arctan (x) + \frac{2 x}{x ^{2} + 1} + \frac{1}{x ^{2} + 1} + C

Graph

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