integral of (x^4-\sqrt[3]{x}+3x)/(5x^3)

Lösung

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

\frac{5 x ^{\frac{14}{3}} + 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}} + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{x ^{4} - 3 x + 3 x}{x ^{3}} d x

Multipliziere aus

\frac{x ^{4} - 3 x + 3 x}{x ^{3}} : x - \frac{1}{x ^{\frac{8}{3}}} + \frac{3}{x ^{2}}

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

= \frac{1}{5} (\int x d x - \int \frac{1}{x ^{\frac{8}{3}}} d x + \int \frac{3}{x ^{2}} d x)

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

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

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

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

Vereinfache

\frac{1}{5} (\frac{x ^{2}}{2} - (- \frac{3}{5 x ^{\frac{5}{3}}}) - \frac{3}{x}) : \frac{5 x ^{\frac{14}{3}} + 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}}

= \frac{5 x ^{\frac{14}{3}} + 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}}

Füge eine Konstante zur Lösung hinzu

= \frac{5 x ^{\frac{14}{3}} + 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}} + C

\int 18 d x

x \to 0 lim (\frac{x ^{5} + 4 x}{x})

d er i v a t i v e f (x) = \frac{1}{3} (x^{2} + 2)^{\frac{3}{2}}

x \to - 1.1 lim (\frac{1}{x ^{2}})

\frac{\partial}{\partial x} (\frac{( 3 x - 2 y )}{3 x + 2 y})