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

Lösung

\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 ∣ + C

Schritte zur Lösung

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

Lange Division

\frac{x ^{5}}{x + 1} : x^{4} - x^{3} + x^{2} - x + 1 - \frac{1}{x + 1}

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

\int x^{4} d x = \frac{x ^{5}}{5}

\int x^{3} d x = \frac{x ^{4}}{4}

\int x^{2} d x = \frac{x ^{3}}{3}

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

\int 1 d x = x

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e \frac{4 x}{x - 5}

d er i v a t i v e y = tan (\frac{t}{1 + t ^{2}})

\int_{0}^{1} 4 π (x^{2} - x^{10}) d x

t an g e n t \frac{3}{x}

\int 2 x (x^{2} + 5)^{5} d x