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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2}{5} (1 + x)^{\frac{5}{2}} - \frac{4}{3} (1 + x)^{\frac{3}{2}} + 2 1 + x + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

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

= \frac{2}{5} u^{\frac{5}{2}} - \frac{4}{3} u^{\frac{3}{2}} + 2 u

Setze in

u = 1 + x

ein

= \frac{2}{5} (1 + x)^{\frac{5}{2}} - \frac{4}{3} (1 + x)^{\frac{3}{2}} + 2 1 + x

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (1 + x)^{\frac{5}{2}} - \frac{4}{3} (1 + x)^{\frac{3}{2}} + 2 1 + x + C

Graph

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