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

Lösung

\int (3 x^{5} - x) (x + 2) d x

\frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2}) + C

Schritte zur Lösung

\int (3 x^{5} - x) (x + 2) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{( 3 u - 5 u ) ( 5 u + 2 )}{u ^{\frac{4}{5}}} : \frac{1}{u ^{\frac{4}{15}}} + \frac{2}{u ^{\frac{7}{15}}} - \frac{1}{u ^{\frac{2}{5}}} - \frac{2}{u ^{\frac{3}{5}}}

= \frac{1}{5} \cdot \int \frac{1}{u ^{\frac{4}{15}}} + \frac{2}{u ^{\frac{7}{15}}} - \frac{1}{u ^{\frac{2}{5}}} - \frac{2}{u ^{\frac{3}{5}}} 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{1}{5} (\int \frac{1}{u ^{\frac{4}{15}}} d u + \int \frac{2}{u ^{\frac{7}{15}}} d u - \int \frac{1}{u ^{\frac{2}{5}}} d u - \int \frac{2}{u ^{\frac{3}{5}}} d u)

\int \frac{1}{u ^{\frac{4}{15}}} d u = \frac{15}{11} u^{\frac{11}{15}}

\int \frac{2}{u ^{\frac{7}{15}}} d u = \frac{15}{4} u^{\frac{8}{15}}

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

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

= \frac{1}{5} (\frac{15}{11} u^{\frac{11}{15}} + \frac{15}{4} u^{\frac{8}{15}} - \frac{5}{3} u^{\frac{3}{5}} - 5 u^{\frac{2}{5}})

Setze in

u = x^{5}

ein

= \frac{1}{5} (\frac{15}{11} (x^{5})^{\frac{11}{15}} + \frac{15}{4} (x^{5})^{\frac{8}{15}} - \frac{5}{3} (x^{5})^{\frac{3}{5}} - 5 (x^{5})^{\frac{2}{5}})

Vereinfache

\frac{1}{5} (\frac{15}{11} (x^{5})^{\frac{11}{15}} + \frac{15}{4} (x^{5})^{\frac{8}{15}} - \frac{5}{3} (x^{5})^{\frac{3}{5}} - 5 (x^{5})^{\frac{2}{5}}) : \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2})

= \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2}) + C

y^{^{'''}} - 8 y = 0

\frac{d}{d x} (ln (2 - 5 x))

\frac{d}{d x} (arctan (x) - x^{2})

\int sin (7 x) ln (sin (7 x)) d x

y^{^{'}} = 2 (1 + y^{2}) x