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Beliebt Rechnen >

integral of zsqrt(1-z)

Lösung

\int z 1 - z d z

Lösung

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

Schritte zur Lösung

\int z 1 - z d z

Wende U-Substitution an

= \int - (- u + 1) u d u

Entferne die Konstante:

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

= - \int (- u + 1) u d u

Multipliziere aus

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

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

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

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

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

Setze in

u = 1 - z

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches 0 of (x-2)/(|x-2|)

x \to 0 lim (\frac{x - 2}{∣ x - 2 ∣})

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\int_{0}^{4} y - y + 2 d y

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\frac{d}{d x} (4 x (x^{2} - 4))

integral of 1/(2x^2-x)

\int \frac{1}{2 x ^{2} - x} d x

derivative f(x)=5x-7

d er i v a t i v e f (x) = 5 x - 7