integral of (x-1/x)^3

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

(x - \frac{1}{x})^{3} : x^{3} - 3 x + \frac{3}{x} - \frac{1}{x ^{3}}

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

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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