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Beliebt Rechnen >

integral of ((x^2-x+2))/(x^3-1)

Lösung

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

Lösung

\frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{3 ( x - 1 )} d x = \frac{2}{3} ln ∣ x - 1 ∣

\int \frac{x - 4}{3 ( x ^{2} + x + 1 )} d x = \frac{1}{6} (ln 4 x^{2} + 4 x + 4 - 63 arctan (\frac{2 x + 1}{3}))

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (ln 4 x^{2} + 4 x + 4 - 63 arctan (\frac{2 x + 1}{3}))

Vereinfache

\frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (ln 4 x^{2} + 4 x + 4 - 63 arctan (\frac{2 x + 1}{3})) : \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Graph

Beliebte Beispiele

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

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

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

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

integral of 1/(4x^3+4x^2+x)

\int \frac{1}{4 x ^{3} + 4 x ^{2} + x} d x

integral of 1/(3cos(x))

\int \frac{1}{3 cos ( x )} d x

integral of 2/5 x^{5/3}+C

\int \frac{2}{5} x^{\frac{5}{3}} + C d x