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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Lange Division

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{1}{x + 2} d x = 2 (x + 2 - 2 ln x + 2)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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