integral of x/(sqrt(3x+2))

Lösung

\int \frac{x}{3 x + 2} d x

\frac{2}{9} (\frac{1}{3} (3 x + 2)^{\frac{3}{2}} - 2 3 x + 2) + C

Schritte zur Lösung

\int \frac{x}{3 x + 2} d x

Wende U-Substitution an

= \int \frac{2 ( u ^{2} - 2 )}{9} d u

Entferne die Konstante:

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

= \frac{2}{9} \cdot \int u^{2} - 2 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

= \frac{2}{9} (\int u^{2} d u - \int 2 d u)

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

\int 2 d u = 2 u

= \frac{2}{9} (\frac{u ^{3}}{3} - 2 u)

Setze in

u = 3 x + 2

ein

= \frac{2}{9} (\frac{( 3 x + 2 ) ^{3}}{3} - 2 3 x + 2)

Vereinfache

\frac{2}{9} (\frac{( 3 x + 2 ) ^{3}}{3} - 2 3 x + 2) : \frac{2}{9} (\frac{1}{3} (3 x + 2)^{\frac{3}{2}} - 2 3 x + 2)

= \frac{2}{9} (\frac{1}{3} (3 x + 2)^{\frac{3}{2}} - 2 3 x + 2)

Füge eine Konstante zur Lösung hinzu

= \frac{2}{9} (\frac{1}{3} (3 x + 2)^{\frac{3}{2}} - 2 3 x + 2) + C

\frac{\partial}{\partial x} (e^{y l n (z)})

d er i v a t i v e \frac{9}{t ^{3}}

\int_{- π}^{π} t sin (n t) d t

k = 0 \sum \infty x^{k + 1}

im pl i c i t \frac{d y}{d x}, x y = 9