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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u - 1}{4 u ^{\frac{1}{2}}} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{u - 1}{u ^{\frac{1}{2}}} d u

Multipliziere aus

\frac{u - 1}{u ^{\frac{1}{2}}} : u^{\frac{1}{2}} - \frac{1}{u ^{\frac{1}{2}}}

= \frac{1}{4} \cdot \int u^{\frac{1}{2}} - \frac{1}{u ^{\frac{1}{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

= \frac{1}{4} (\int u^{\frac{1}{2}} d u - \int \frac{1}{u ^{\frac{1}{2}}} d u)

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{1}{u ^{\frac{1}{2}}} d u = 2 u^{\frac{1}{2}}

= \frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} - 2 u^{\frac{1}{2}})

Setze in

u = 2 x + 1

ein

= \frac{1}{4} (\frac{2}{3} (2 x + 1)^{\frac{3}{2}} - 2 (2 x + 1)^{\frac{1}{2}})

Vereinfache

\frac{1}{4} (\frac{2}{3} (2 x + 1)^{\frac{3}{2}} - 2 (2 x + 1)^{\frac{1}{2}}) : \frac{1}{3} (x - 1) (2 x + 1)^{\frac{1}{2}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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