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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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