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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 1 - 2 arctan (x) - \frac{1}{2} ln 4 x^{2} - 4 x + 4 + \frac{5}{3} arctan (\frac{2 x - 1}{3}) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{2 x + 1}{( 1 + x ^{2} ) ( 1 - x + x ^{2} )} : \frac{x - 2}{x ^{2} + 1} + \frac{- x + 3}{x ^{2} - x + 1}

= \int \frac{x - 2}{x ^{2} + 1} + \frac{- x + 3}{x ^{2} - x + 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{x - 2}{x ^{2} + 1} d x + \int \frac{- x + 3}{x ^{2} - x + 1} d x

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

\int \frac{- x + 3}{x ^{2} - x + 1} d x = - \frac{1}{2} ln 4 x^{2} - 4 x + 4 + \frac{5}{3} arctan (\frac{2 x - 1}{3})

= \frac{1}{2} ln x^{2} + 1 - 2 arctan (x) - \frac{1}{2} ln 4 x^{2} - 4 x + 4 + \frac{5}{3} arctan (\frac{2 x - 1}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 1 - 2 arctan (x) - \frac{1}{2} ln 4 x^{2} - 4 x + 4 + \frac{5}{3} arctan (\frac{2 x - 1}{3}) + C

Graph

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