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Beliebt Rechnen >

integral of (x^2+x+8)sqrt(x+7)

Lösung

\int (x^{2} + x + 8) x + 7 d x

Lösung

\frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + \frac{100}{3} (x + 7)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x^{2} + x + 8) x + 7 d x

Multipliziere aus

(x^{2} + x + 8) x + 7 : x^{2} x + 7 + x x + 7 + 8 x + 7

= \int x^{2} x + 7 + x x + 7 + 8 x + 7 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 + 7 d x + \int x x + 7 d x + \int 8 x + 7 d x

\int x^{2} x + 7 d x = \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}}

\int x x + 7 d x = \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}}

\int 8 x + 7 d x = \frac{16}{3} (x + 7)^{\frac{3}{2}}

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}} + \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + \frac{16}{3} (x + 7)^{\frac{3}{2}}

Vereinfache

\frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}} + \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + \frac{16}{3} (x + 7)^{\frac{3}{2}} : \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + \frac{100}{3} (x + 7)^{\frac{3}{2}}

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + \frac{100}{3} (x + 7)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + \frac{100}{3} (x + 7)^{\frac{3}{2}} + C

Graph

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