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

Lösung

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

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

Schritte zur Lösung

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

Lange Division

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int 2 e^{- 2 t} + 4 t + 1 d t

\int - x sin (\frac{x}{2}) cos (\frac{x}{2}) d x

\int \frac{( 1 - r ^{2} ) ^{\frac{5}{2}}}{r ^{8}} d r

\int x - x^{2} + 1 d x

\int \frac{2 ( 3 + 5 x ) ^{5}}{5} d x