integral of x^9sqrt(x^5+7)

Lösung

\int x^{9} x^{5} + 7 d x

\frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x^{9} x^{5} + 7 d x

Wende U-Substitution an

= \int \frac{2 ( u ^{2} - 7 ) 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} - 7) u^{2} d u

Multipliziere aus

(u^{2} - 7) u^{2} : u^{4} - 7 u^{2}

= \frac{2}{5} \cdot \int u^{4} - 7 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 7 u^{2} d u)

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

\int 7 u^{2} d u = \frac{7 u ^{3}}{3}

= \frac{2}{5} (\frac{u ^{5}}{5} - \frac{7 u ^{3}}{3})

Setze in

u = x^{5} + 7

ein

= \frac{2}{5} (\frac{( x ^{5} + 7 ) ^{5}}{5} - \frac{7 ( x ^{5} + 7 ) ^{3}}{3})

Vereinfache

\frac{2}{5} (\frac{( x ^{5} + 7 ) ^{5}}{5} - \frac{7 ( x ^{5} + 7 ) ^{3}}{3}) : \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}})

= \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}}) + C

d er i v a t i v e \frac{x ^{2} - 25}{x - 5}

y^{^{'}} = \frac{y ^{2}}{x}

\int x \cdot 1 - x d x

x y^{^{'}} - y = x^{2} - x + 2

s \to 5 lim (\frac{5}{( s - 5 ) ^{2}})