integral of x^5sqrt(x^3+4)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int (u - 4) u d u

Multipliziere aus

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

= \frac{1}{3} \cdot \int u^{\frac{3}{2}} - 4 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

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

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

\int 4 u d u = \frac{8}{3} u^{\frac{3}{2}}

= \frac{1}{3} (\frac{2}{5} u^{\frac{5}{2}} - \frac{8}{3} u^{\frac{3}{2}})

Setze in

u = x^{3} + 4

ein

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

Füge eine Konstante zur Lösung hinzu

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

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