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

Lösung

\int \frac{x ^{3}}{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})) + x + C

Schritte zur Lösung

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

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

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

\int \frac{1}{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}))

\int 1 d x = 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}))) + x

Vereinfache

- (\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}))) + 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})) + 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})) + x

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})) + x + C

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

\frac{d q}{d t} = 5 - 5 q

d er i v a t i v e y = lo g_{2} (x) \cdot lo g_{5} (x)

x \to 2 - lim (\frac{1}{x ^{2} - 4})

in v erse l a pl a ce \frac{e ^{- 4 s}}{( s + 2 ) ^{3}}