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Beliebt Rechnen >

integral of x\sqrt[3]{16-3x}

Lösung

\int x 3 16 - 3 x d x

Lösung

\frac{1}{21} ((- 3 x + 16)^{\frac{7}{3}} - 28 (- 3 x + 16)^{\frac{4}{3}}) + C

Schritte zur Lösung

\int x 3 16 - 3 x d x

Wende U-Substitution an

= \int \frac{3 u ( u - 16 )}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int 3 u (u - 16) d u

Multipliziere aus

3 u (u - 16) : u^{\frac{4}{3}} - 16 3 u

= \frac{1}{9} \cdot \int u^{\frac{4}{3}} - 16 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

= \frac{1}{9} (\int u^{\frac{4}{3}} d u - \int 16 3 u d u)

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

\int 16 3 u d u = 12 u^{\frac{4}{3}}

= \frac{1}{9} (\frac{3}{7} u^{\frac{7}{3}} - 12 u^{\frac{4}{3}})

Setze in

u = 16 - 3 x

ein

= \frac{1}{9} (\frac{3}{7} (16 - 3 x)^{\frac{7}{3}} - 12 (16 - 3 x)^{\frac{4}{3}})

Vereinfache

\frac{1}{9} (\frac{3}{7} (16 - 3 x)^{\frac{7}{3}} - 12 (16 - 3 x)^{\frac{4}{3}}) : \frac{1}{21} ((- 3 x + 16)^{\frac{7}{3}} - 28 (- 3 x + 16)^{\frac{4}{3}})

= \frac{1}{21} ((- 3 x + 16)^{\frac{7}{3}} - 28 (- 3 x + 16)^{\frac{4}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{21} ((- 3 x + 16)^{\frac{7}{3}} - 28 (- 3 x + 16)^{\frac{4}{3}}) + C

Graph

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