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Beliebt Rechnen >

integral of (x^2+x+9)sqrt(x+8)

Lösung

\int (x^{2} + x + 9) x + 8 d x

Lösung

\frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x^{2} + x + 9) x + 8 d x

Multipliziere aus

(x^{2} + x + 9) x + 8 : x^{2} x + 8 + x x + 8 + 9 x + 8

= \int x^{2} x + 8 + x x + 8 + 9 x + 8 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} x + 8 d x + \int x x + 8 d x + \int 9 x + 8 d x

\int x^{2} x + 8 d x = \frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}}

\int x x + 8 d x = \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}}

\int 9 x + 8 d x = 6 (x + 8)^{\frac{3}{2}}

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}} + \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}} + 6 (x + 8)^{\frac{3}{2}}

Vereinfache

\frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}} + \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}} + 6 (x + 8)^{\frac{3}{2}} : \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}}

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}} + C

Graph

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