integral of sqrt(x)((-3)/(x^3)-x^2+x/4)

Lösung

\int x (\frac{- 3}{x ^{3}} - x^{2} + \frac{x}{4}) d x

\frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

Schritte zur Lösung

\int x (\frac{- 3}{x ^{3}} - x^{2} + \frac{x}{4}) d x

Multipliziere aus

x (\frac{- 3}{x ^{3}} - x^{2} + \frac{x}{4}) : - \frac{3}{x ^{\frac{5}{2}}} + \frac{x ^{\frac{3}{2}}}{4} - x^{\frac{5}{2}}

= \int - \frac{3}{x ^{\frac{5}{2}}} + \frac{x ^{\frac{3}{2}}}{4} - x^{\frac{5}{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 \frac{3}{x ^{\frac{5}{2}}} d x + \int \frac{x ^{\frac{3}{2}}}{4} d x - \int x^{\frac{5}{2}} d x

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

\int \frac{x ^{\frac{3}{2}}}{4} d x = \frac{1}{10} x^{\frac{5}{2}}

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= - (- \frac{2}{x ^{\frac{3}{2}}}) + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}}

Vereinfache

= \frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C