integral of x^{11}sqrt(x^6+5)

Lösung

\int x^{11} x^{6} + 5 d x

\frac{1}{45} (3 x^{6} - 10) (x^{6} + 5)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{11} x^{6} + 5 d x

Wende U-Substitution an

= \int \frac{( u - 5 ) u}{6} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int (u - 5) u d u

Multipliziere aus

(u - 5) u : u^{\frac{3}{2}} - 5 u

= \frac{1}{6} \cdot \int u^{\frac{3}{2}} - 5 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}{6} (\int u^{\frac{3}{2}} d u - \int 5 u d u)

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

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= \frac{1}{6} (\frac{2}{5} u^{\frac{5}{2}} - \frac{10}{3} u^{\frac{3}{2}})

Setze in

u = x^{6} + 5

ein

= \frac{1}{6} (\frac{2}{5} (x^{6} + 5)^{\frac{5}{2}} - \frac{10}{3} (x^{6} + 5)^{\frac{3}{2}})

Vereinfache

\frac{1}{6} (\frac{2}{5} (x^{6} + 5)^{\frac{5}{2}} - \frac{10}{3} (x^{6} + 5)^{\frac{3}{2}}) : \frac{1}{45} (3 x^{6} - 10) (x^{6} + 5)^{\frac{3}{2}}

= \frac{1}{45} (3 x^{6} - 10) (x^{6} + 5)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{45} (3 x^{6} - 10) (x^{6} + 5)^{\frac{3}{2}} + C

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