integral of xsqrt(9-x)

Lösung

\int x 9 - x d x

\frac{2}{5} (9 - x)^{\frac{5}{2}} - 6 (9 - x)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x 9 - x d x

Wende U-Substitution an

= \int - (- u + 9) u d u

Entferne die Konstante:

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

= - \int (- u + 9) u d u

Multipliziere aus

(- u + 9) u : - u^{\frac{3}{2}} + 9 u

= - \int - u^{\frac{3}{2}} + 9 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

= - (- \int u^{\frac{3}{2}} d u + \int 9 u d u)

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

\int 9 u d u = 6 u^{\frac{3}{2}}

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

Setze in

u = 9 - x

ein

= - (- \frac{2}{5} (9 - x)^{\frac{5}{2}} + 6 (9 - x)^{\frac{3}{2}})

Vereinfache

= \frac{2}{5} (9 - x)^{\frac{5}{2}} - 6 (9 - x)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (9 - x)^{\frac{5}{2}} - 6 (9 - x)^{\frac{3}{2}} + C

d er i v a t i v e \frac{y ^{2} + x ^{2}}{y ^{2} - x ^{2}}

t an g e n t f (x) = \frac{2}{x}, at x = 1

t an g e n t x^{2} - 3 x + 3

\int 2 y e^{2 z} d z

d er i v a t i v e h (g^{^{'}} (x))