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Beliebt Rechnen >

integral of (sqrt(x))/(sqrt(x)-2)

Lösung

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

Lösung

x + 4 x + 8 ln x - 2 - 12 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2 ( u ^{2} + 4 u + 4 )}{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} + 4 u + 4}{u} d u

Multipliziere aus

\frac{u ^{2} + 4 u + 4}{u} : u + 4 + \frac{4}{u}

= 2 \cdot \int u + 4 + \frac{4}{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

= 2 (\int u d u + \int 4 d u + \int \frac{4}{u} d u)

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

\int 4 d u = 4 u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} + 4 u + 4 ln ∣ u ∣)

Setze in

u = x - 2

ein

= 2 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln x - 2)

Vereinfache

2 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln x - 2) : x + 4 x + 8 ln x - 2 - 12

= x + 4 x + 8 ln x - 2 - 12

Füge eine Konstante zur Lösung hinzu

= x + 4 x + 8 ln x - 2 - 12 + C

Graph

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