integral of 3/(sqrt(x+3)+\sqrt{x)}

Lösung

\int \frac{3}{x + 3 + x} d x

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

Schritte zur Lösung

\int \frac{3}{x + 3 + x} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{1}{x + 3 + x} d x

Multipliziere mit der Konjugation von

x + 3 + x : \frac{3}{x + 3 - x}

= 3 \cdot \int \frac{1}{\frac{3}{x + 3 - x}} d x

Entferne die Konstante:

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

= 3 \cdot \frac{1}{3} \cdot \int \frac{1}{\frac{1}{x + 3 - x}} d x

Vereinfache

= 3 \cdot \frac{1}{3} \cdot \int x + 3 - 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

= 3 \cdot \frac{1}{3} (\int x + 3 d x - \int x d x)

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int x + \frac{1}{x} - 7 d x

x \to 7 lim (5 x^{2} - 5 x + 6)

x \to 0 lim (\frac{1}{1 + cos ( x )})

\frac{d}{d x} (\frac{3}{4} (x^{2} - 1)^{\frac{2}{3}})

\int 7 sin^{3} (x) cos^{2} (x) d x