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Beliebt Rechnen >

integral of (4-4x)/(1-sqrt(x))

Lösung

\int \frac{4 - 4 x}{1 - x} d x

Lösung

8 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2}) + C

Schritte zur Lösung

\int \frac{4 - 4 x}{1 - x} d x

Wende U-Substitution an

= \int 8 u (u + 1) d u

Entferne die Konstante:

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

= 8 \cdot \int u (u + 1) d u

Multipliziere aus

u (u + 1) : u^{2} + u

= 8 \cdot \int u^{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

= 8 (\int u^{2} d u + \int u d u)

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

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

= 8 (\frac{u ^{3}}{3} + \frac{u ^{2}}{2})

Setze in

u = x

ein

= 8 (\frac{( x ) ^{3}}{3} + \frac{( x ) ^{2}}{2})

Vereinfache

8 (\frac{( x ) ^{3}}{3} + \frac{( x ) ^{2}}{2}) : 8 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2})

= 8 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2})

Füge eine Konstante zur Lösung hinzu

= 8 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2}) + C

Graph

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