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Beliebt Rechnen >

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

Lösung

\int (x^{2} - 1) 2 x + 1 d x

Lösung

\frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{11}{35} (2 x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x^{2} - 1) 2 x + 1 d x

Multipliziere aus

(x^{2} - 1) 2 x + 1 : x^{2} 2 x + 1 - 2 x + 1

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 2 x + 1 d x = \frac{1}{3} (2 x + 1)^{\frac{3}{2}}

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

Vereinfache

\frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} - \frac{1}{3} (2 x + 1)^{\frac{3}{2}} : \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{11}{35} (2 x + 1)^{\frac{3}{2}}

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{11}{35} (2 x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{11}{35} (2 x + 1)^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

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