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Beliebt Rechnen >

integral of (sqrt(x))/(sqrt(1+\sqrt{x))}

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= 4 \cdot \int u^{4} - 2 u^{2} + 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

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

\int u^{4} d u = \frac{u ^{5}}{5}

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

\int 1 d u = u

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

Setze in

u = 1 + x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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