integral of xsqrt(3x+1)

Lösung

\int x 3 x + 1 d x

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

Schritte zur Lösung

\int x 3 x + 1 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 + 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

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