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Beliebt Rechnen >

integral of x^2*sqrt(2x+1)

Lösung

\int x^{2} \cdot 2 x + 1 d x

Lösung

\frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{2} 2 x + 1 d x

Wende U-Substitution an

= \int \frac{u ( u - 1 ) ^{2}}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int u (u - 1)^{2} d u

Multipliziere aus

u (u - 1)^{2} : u^{\frac{5}{2}} - 2 u^{\frac{3}{2}} + u

= \frac{1}{8} \cdot \int u^{\frac{5}{2}} - 2 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}{8} (\int u^{\frac{5}{2}} d u - \int 2 u^{\frac{3}{2}} d u + \int u d u)

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

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

= \frac{1}{8} (\frac{2}{7} u^{\frac{7}{2}} - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 2 x + 1

ein

= \frac{1}{8} (\frac{2}{7} (2 x + 1)^{\frac{7}{2}} - \frac{4}{5} (2 x + 1)^{\frac{5}{2}} + \frac{2}{3} (2 x + 1)^{\frac{3}{2}})

Vereinfache

\frac{1}{8} (\frac{2}{7} (2 x + 1)^{\frac{7}{2}} - \frac{4}{5} (2 x + 1)^{\frac{5}{2}} + \frac{2}{3} (2 x + 1)^{\frac{3}{2}}) : \frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}}

= \frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} + C

Graph

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