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Beliebt Rechnen >

integral of x^5\sqrt[3]{x^2+1}

Lösung

\int x^{5} 3 x^{2} + 1 d x

Lösung

\frac{1}{2} (\frac{3}{10} (x^{2} + 1)^{\frac{10}{3}} - \frac{6}{7} (x^{2} + 1)^{\frac{7}{3}} + \frac{3}{4} (x^{2} + 1)^{\frac{4}{3}}) + C

Schritte zur Lösung

\int x^{5} 3 x^{2} + 1 d x

Wende U-Substitution an

= \int \frac{u ^{2} 3 u + 1}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u^{2} 3 u + 1 d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int (v - 1)^{2} 3 v d v

Multipliziere aus

(v - 1)^{2} 3 v : v^{\frac{7}{3}} - 2 v^{\frac{4}{3}} + 3 v

= \frac{1}{2} \cdot \int v^{\frac{7}{3}} - 2 v^{\frac{4}{3}} + 3 v d v

Wende die Summenregel an:

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

= \frac{1}{2} (\int v^{\frac{7}{3}} d v - \int 2 v^{\frac{4}{3}} d v + \int 3 v d v)

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

\int 2 v^{\frac{4}{3}} d v = \frac{6}{7} v^{\frac{7}{3}}

\int 3 v d v = \frac{3}{4} v^{\frac{4}{3}}

= \frac{1}{2} (\frac{3}{10} v^{\frac{10}{3}} - \frac{6}{7} v^{\frac{7}{3}} + \frac{3}{4} v^{\frac{4}{3}})

Ersetze zurück

= \frac{1}{2} (\frac{3}{10} (x^{2} + 1)^{\frac{10}{3}} - \frac{6}{7} (x^{2} + 1)^{\frac{7}{3}} + \frac{3}{4} (x^{2} + 1)^{\frac{4}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{3}{10} (x^{2} + 1)^{\frac{10}{3}} - \frac{6}{7} (x^{2} + 1)^{\frac{7}{3}} + \frac{3}{4} (x^{2} + 1)^{\frac{4}{3}}) + C

Graph

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