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Beliebt Rechnen >

integral of (sqrt(w))/(sqrt(1-\sqrt{w))}

Lösung

\int \frac{w}{1 - w} d w

Lösung

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

Schritte zur Lösung

\int \frac{w}{1 - w} d w

Wende U-Substitution an

= \int \frac{2 u ^{2}}{1 - u} d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{u ^{2}}{1 - u} d u

Wende U-Substitution an

= 2 \cdot \int - 2 (- v^{2} + 1)^{2} d v

Entferne die Konstante:

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

= 2 (- 2 \cdot \int (- v^{2} + 1)^{2} d v)

Multipliziere aus

(- v^{2} + 1)^{2} : v^{4} - 2 v^{2} + 1

= 2 (- 2 \cdot \int v^{4} - 2 v^{2} + 1 d v)

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (- 2 (\int v^{4} d v - \int 2 v^{2} d v + \int 1 d v))

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

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

\int 1 d v = v

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

Ersetze zurück

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

Vereinfache

2 - 2 \frac{( 1 - w ) ^{5}}{5} - \frac{2 ( 1 - w ) ^{3}}{3} + 1 - w : - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w)

= - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w)

Füge eine Konstante zur Lösung hinzu

= - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w) + C

Graph

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