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

Lösung

\int \frac{x}{3 x + 4} d x

\frac{3}{5} (x + 4)^{\frac{5}{3}} - 6 (x + 4)^{\frac{2}{3}} + C

Schritte zur Lösung

\int \frac{x}{3 x + 4} d x

Wende U-Substitution an

= \int \frac{u - 4}{3 u} d u

Multipliziere aus

\frac{u - 4}{3 u} : u^{\frac{2}{3}} - \frac{4}{3 u}

= \int u^{\frac{2}{3}} - \frac{4}{3 u} 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

= \int u^{\frac{2}{3}} d u - \int \frac{4}{3 u} d u

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{4}{3 u} d u = 6 u^{\frac{2}{3}}

= \frac{3}{5} u^{\frac{5}{3}} - 6 u^{\frac{2}{3}}

Setze in

u = x + 4

ein

= \frac{3}{5} (x + 4)^{\frac{5}{3}} - 6 (x + 4)^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{5} (x + 4)^{\frac{5}{3}} - 6 (x + 4)^{\frac{2}{3}} + C

x \to 3 lim (\frac{1}{( x - 3 ) ^{8}})

x \to - 3 lim (\frac{x ^{2} - 9}{3 + x})

\frac{d}{d x} (\frac{2}{3 x ^{\frac{1}{3}}})

\int_{3}^{5} \frac{x}{x ^{2} - 3} d x

d er i v a t i v e (6 x^{2} - 10 x) e^{x}