integral of xsqrt(5-x)

Lösung

\int x 5 - x d x

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

Schritte zur Lösung

\int x 5 - x d x

Wende U-Substitution an

= \int - (- u + 5) u d u

Entferne die Konstante:

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

= - \int (- u + 5) u d u

Multipliziere aus

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

= - \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

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

Setze in

u = 5 - x

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = ln (4^{x} + 8^{x})

\frac{d y}{d x} + 3 x^{2} y = 9 x^{2}

d er i v a t i v e - \frac{1}{x ^{2} ( sec ^{2} ( y ) )}

\int e^{2 x - 3} d x

x^{2} y^{^{'}} + 2 x y = 9 cos^{2} (x)