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Beliebt Rechnen >

integral of x(x+2)^{1/2}

Lösung

\int x (x + 2)^{\frac{1}{2}} d x

Lösung

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

Schritte zur Lösung

\int x (x + 2)^{\frac{1}{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

(u^{2} - 2) u^{2} : u^{4} - 2 u^{2}

= 2 \cdot \int u^{4} - 2 u^{2} 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^{4} d u - \int 2 u^{2} d u)

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

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

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

Setze in

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

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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