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Beliebt Rechnen >

integral of (x+1)(sqrt(3x+2))

Lösung

\int (x + 1) (3 x + 2) d x

Lösung

\frac{1}{9} (\frac{2}{5} (3 x + 2)^{\frac{5}{2}} + \frac{2}{3} (3 x + 2)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int (x + 1) 3 x + 2 d x

Wende U-Substitution an

= \int \frac{u ( u + 1 )}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int u (u + 1) d u

Multipliziere aus

u (u + 1) : u^{\frac{3}{2}} + u

= \frac{1}{9} \cdot \int u^{\frac{3}{2}} + u 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{1}{9} (\int u^{\frac{3}{2}} d u + \int u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

= \frac{1}{9} (\frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 3 x + 2

ein

= \frac{1}{9} (\frac{2}{5} (3 x + 2)^{\frac{5}{2}} + \frac{2}{3} (3 x + 2)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (\frac{2}{5} (3 x + 2)^{\frac{5}{2}} + \frac{2}{3} (3 x + 2)^{\frac{3}{2}}) + C

Graph

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