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Beliebt Rechnen >

integral of x/(sqrt(x+8))

Lösung

\int \frac{x}{x + 8} d x

Lösung

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

Schritte zur Lösung

\int \frac{x}{x + 8} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 2 \cdot \int u^{2} - 8 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^{2} d u - \int 8 d u)

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

\int 8 d u = 8 u

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

Setze in

u = x + 8

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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