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Beliebt Rechnen >

integral of x/(1+sqrt(2x+1))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

\frac{u - 1}{1 + u} = u - 1

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

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

\int 1 d u = u

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

Setze in

u = 2 x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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