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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

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

= \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

= \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{2}{3} u^{\frac{3}{2}} - u

Setze in

u = 1 + x

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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