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Beliebt Rechnen >

integral of 5/(sqrt(x+5)+\sqrt{x)}

Lösung

\int \frac{5}{x + 5 + x} d x

Lösung

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

Schritte zur Lösung

\int \frac{5}{x + 5 + x} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{1}{x + 5 + x} d x

Multipliziere mit der Konjugation von

x + 5 + x : \frac{5}{x + 5 - x}

= 5 \cdot \int \frac{1}{\frac{5}{x + 5 - x}} d x

Entferne die Konstante:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{\frac{1}{x + 5 - x}} d x

Vereinfache

= 5 \cdot \frac{1}{5} \cdot \int x + 5 - x 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

= 5 \cdot \frac{1}{5} (\int x + 5 d x - \int x d x)

\int x + 5 d x = \frac{2}{3} (x + 5)^{\frac{3}{2}}

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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