integral of (5x^2-10x)/(\sqrt[4]{3x^3)}

Lösung

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

\frac{20}{9 4 3} x^{\frac{9}{4}} - \frac{8}{4 3} x^{\frac{5}{4}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{5 ( 3 u - 2 )}{3 4 3 u ^{\frac{7}{12}}} d u

Entferne die Konstante:

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

= \frac{5}{3 4 3} \cdot \int \frac{3 u - 2}{u ^{\frac{7}{12}}} d u

Multipliziere aus

\frac{3 u - 2}{u ^{\frac{7}{12}}} : \frac{1}{4 u} - \frac{2}{u ^{\frac{7}{12}}}

= \frac{5}{3 4 3} \cdot \int \frac{1}{4 u} - \frac{2}{u ^{\frac{7}{12}}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{5}{3 4 3} (\int \frac{1}{4 u} d u - \int \frac{2}{u ^{\frac{7}{12}}} d u)

\int \frac{1}{4 u} d u = \frac{4}{3} u^{\frac{3}{4}}

\int \frac{2}{u ^{\frac{7}{12}}} d u = \frac{24}{5} u^{\frac{5}{12}}

= \frac{5}{3 4 3} (\frac{4}{3} u^{\frac{3}{4}} - \frac{24}{5} u^{\frac{5}{12}})

Setze in

u = x^{3}

ein

= \frac{5}{3 4 3} (\frac{4}{3} (x^{3})^{\frac{3}{4}} - \frac{24}{5} (x^{3})^{\frac{5}{12}})

Vereinfache

\frac{5}{3 4 3} (\frac{4}{3} (x^{3})^{\frac{3}{4}} - \frac{24}{5} (x^{3})^{\frac{5}{12}}) : \frac{20}{9 4 3} x^{\frac{9}{4}} - \frac{8}{4 3} x^{\frac{5}{4}}

= \frac{20}{9 4 3} x^{\frac{9}{4}} - \frac{8}{4 3} x^{\frac{5}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{20}{9 4 3} x^{\frac{9}{4}} - \frac{8}{4 3} x^{\frac{5}{4}} + C

d er i v a t i v e 1 - 49 x^{2} arccos (7 x)

x \to 2 lim (7 (x) + 10)

f (x) = x^{\frac{4}{5}}

\int cos^{4} (7 x + 9) d x

\int_{- \infty}^{0} e^{8 x} d x