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Beliebt Rechnen >

integral of x(x+6)^{1/2}

Lösung

\int x (x + 6)^{\frac{1}{2}} d x

Lösung

2 (\frac{1}{5} (x + 6)^{\frac{5}{2}} - 2 (x + 6)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x (x + 6)^{\frac{1}{2}} d x

Wende U-Substitution an

= \int 2 (u^{2} - 6) 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} - 6) u^{2} d u

Multipliziere aus

(u^{2} - 6) u^{2} : u^{4} - 6 u^{2}

= 2 \cdot \int u^{4} - 6 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 6 u^{2} d u)

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

\int 6 u^{2} d u = 2 u^{3}

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

Setze in

u = (x + 6)^{\frac{1}{2}}

ein

= 2 \frac{( ( x + 6 ) ^{\frac{1}{2}} ) ^{5}}{5} - 2 ((x + 6)^{\frac{1}{2}})^{3}

Vereinfache

2 \frac{( ( x + 6 ) ^{\frac{1}{2}} ) ^{5}}{5} - 2 ((x + 6)^{\frac{1}{2}})^{3} : 2 (\frac{1}{5} (x + 6)^{\frac{5}{2}} - 2 (x + 6)^{\frac{3}{2}})

= 2 (\frac{1}{5} (x + 6)^{\frac{5}{2}} - 2 (x + 6)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{5} (x + 6)^{\frac{5}{2}} - 2 (x + 6)^{\frac{3}{2}}) + C

Graph

Beliebte Beispiele

integral of 1/(5x+2)

\int \frac{1}{5 x + 2} d x

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x \to 2 lim (\frac{1}{x - 1})

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