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Beliebt Rechnen >

integral of (x^2+x)\sqrt[3]{x+7}

Lösung

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

Lösung

\frac{3}{10} (x + 7)^{\frac{10}{3}} - \frac{39}{7} (x + 7)^{\frac{7}{3}} + \frac{63}{2} (x + 7)^{\frac{4}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int 3 u (u^{2} - 13 u + 42) d u

Multipliziere aus

3 u (u^{2} - 13 u + 42) : u^{\frac{7}{3}} - 13 u^{\frac{4}{3}} + 42 3 u

= \int u^{\frac{7}{3}} - 13 u^{\frac{4}{3}} + 42 3 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

= \int u^{\frac{7}{3}} d u - \int 13 u^{\frac{4}{3}} d u + \int 42 3 u d u

\int u^{\frac{7}{3}} d u = \frac{3}{10} u^{\frac{10}{3}}

\int 13 u^{\frac{4}{3}} d u = \frac{39}{7} u^{\frac{7}{3}}

\int 42 3 u d u = \frac{63}{2} u^{\frac{4}{3}}

= \frac{3}{10} u^{\frac{10}{3}} - \frac{39}{7} u^{\frac{7}{3}} + \frac{63}{2} u^{\frac{4}{3}}

Setze in

u = x + 7

ein

= \frac{3}{10} (x + 7)^{\frac{10}{3}} - \frac{39}{7} (x + 7)^{\frac{7}{3}} + \frac{63}{2} (x + 7)^{\frac{4}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{10} (x + 7)^{\frac{10}{3}} - \frac{39}{7} (x + 7)^{\frac{7}{3}} + \frac{63}{2} (x + 7)^{\frac{4}{3}} + C

Graph

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