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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x^5-x^{6/5}-9x+2)/(x^{1/3)}

\int \frac{x ^{5} - x ^{\frac{6}{5}} - 9 x + 2}{x ^{\frac{1}{3}}} d x

integral of 1/(x^2-10x+29)

\int \frac{1}{x ^{2} - 10 x + 29} d x

integral of-x*ln(4+x^2)

\int - x \cdot ln (4 + x^{2}) d x

integral of 1/2 sin(-4x)

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

integral of 6/((4x-2)(x+1))

\int \frac{6}{( 4 x - 2 ) ( x + 1 )} d x