integral of (x^3+x^2-x+1)/(x^3)

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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