integral of x^9sqrt(x^5+3)

Lösung

\int x^{9} x^{5} + 3 d x

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

Schritte zur Lösung

\int x^{9} x^{5} + 3 d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

\int u^{4} d u = \frac{u ^{5}}{5}

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

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

Setze in

u = x^{5} + 3

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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