integral of (x^4)/(x^3-1)

Lösung

\int \frac{x ^{4}}{x ^{3} - 1} d x

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

Lange Division

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

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

\int x d x = \frac{x ^{2}}{2}

\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}))

= \frac{x ^{2}}{2} + \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{x ^{2}}{2} + \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

\int_{- π}^{π} sin (n x) d x

l a pl a ce t r an s f or m e^{t} (t + 5)^{3}

\frac{\partial ^{2}}{\partial h ( w ) \partial w} (\frac{w}{h ( w ) ( w ) ^{2}})

in v erse l a pl a ce \frac{5 e ^{- s}}{4 s ^{2} + 36}

d er i v a t i v e - 2 cos (\frac{x}{2})