integral of (x^2-3x+\sqrt[3]{x})/(x^3)

Lösung

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

ln ∣ x ∣ + \frac{3}{x} - \frac{3}{5 x ^{\frac{5}{3}}} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} - 3 x + 3 x}{x ^{3}} : \frac{1}{x} - \frac{3}{x ^{2}} + \frac{1}{x ^{\frac{8}{3}}}

= \int \frac{1}{x} - \frac{3}{x ^{2}} + \frac{1}{x ^{\frac{8}{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 \frac{1}{x} d x - \int \frac{3}{x ^{2}} d x + \int \frac{1}{x ^{\frac{8}{3}}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{1}{x ^{\frac{8}{3}}} d x = - \frac{3}{5 x ^{\frac{5}{3}}}

= ln ∣ x ∣ - (- \frac{3}{x}) - \frac{3}{5 x ^{\frac{5}{3}}}

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int x (x^{2} + 5)^{9} d x

\int \frac{cos ^{3} ( 5 x )}{sin ^{4} ( 5 x )} d x

\int \frac{34}{x ^{2} x ^{2} + 49} d x

\int \frac{x ^{2} - 4 x + 1}{x ^{2} + 9} d x

\int cot^{2} (a) x d x