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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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