integral of (6x^5-sqrt(x)+5x^2)/(x^3)

Lösung

\int \frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} d x

2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} d x

Multipliziere aus

\frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} : 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{x}

= \int 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{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 6 x^{2} d x - \int \frac{1}{x ^{\frac{5}{2}}} d x + \int \frac{5}{x} d x

\int 6 x^{2} d x = 2 x^{3}

\int \frac{1}{x ^{\frac{5}{2}}} d x = - \frac{2}{3 x ^{\frac{3}{2}}}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= 2 x^{3} - (- \frac{2}{3 x ^{\frac{3}{2}}}) + 5 ln ∣ x ∣

Vereinfache

= 2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣ + C