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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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