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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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