integral of (x^5+x)/(x+1)

Lösung

\int \frac{x ^{5} + x}{x + 1} d x

\frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

Schritte zur Lösung

\int \frac{x ^{5} + x}{x + 1} d x

Multipliziere aus

\frac{x ^{5} + x}{x + 1} : \frac{x ^{5}}{x + 1} + \frac{x}{x + 1}

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{x ^{5}}{x + 1} d x + \int \frac{x}{x + 1} d x

\int \frac{x ^{5}}{x + 1} d x = \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣

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

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣ + x + 1 - ln ∣ x + 1 ∣

Vereinfache

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

d er i v a t i v e x + 8 x

x \to \infty lim (\frac{e ^{4 x}}{5 x})

d er i v a t i v e (sin (x))^{(87)}

\frac{d}{d x} (\frac{1}{3} e^{- 3 x})

in v erse l a pl a ce \frac{1}{( s ^{2} + 2 s + 2 ) ( s ^{2} + 9 )}