integral of (x^3-4x-1)/(x^2)

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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

\int \frac{2 x ^{5}}{x ^{6} + 5} d x

\int e^{3 + 3 y} d y

\int \frac{5}{6} d x

\int \frac{x ^{2}}{( x ^{2} - 9 ) ^{2}} d x