integral of x/(x^3-1)

Lösung

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

\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{- x + 1}{3 ( x ^{2} + x + 1 )} d x = \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))

= \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

\frac{d}{d x} (ln (x - 1) + cos (x - 1))

\int_{- 1}^{1} 1 - x^{2} d x

\int (4 - 3 x)^{7} d x

l a pl a ce t r an s f or m cos (2 (t - \frac{π}{2}))

\int - 6 sin (x) d x