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Beliebt Rechnen >

integral of (x^2+x+7)sqrt(x+3)

Lösung

\int (x^{2} + x + 7) x + 3 d x

Lösung

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

Schritte zur Lösung

\int (x^{2} + x + 7) x + 3 d x

Multipliziere aus

(x^{2} + x + 7) x + 3 : x^{2} x + 3 + x x + 3 + 7 x + 3

= \int x^{2} x + 3 + x x + 3 + 7 x + 3 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 + 3 d x + \int x x + 3 d x + \int 7 x + 3 d x

\int x^{2} x + 3 d x = \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}}

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

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

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}} + \frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + \frac{14}{3} (x + 3)^{\frac{3}{2}}

Vereinfache

\frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}} + \frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + \frac{14}{3} (x + 3)^{\frac{3}{2}} : \frac{2}{7} (x + 3)^{\frac{7}{2}} - 2 (x + 3)^{\frac{5}{2}} + \frac{26}{3} (x + 3)^{\frac{3}{2}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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