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Beliebt Rechnen >

integral of (1372x)\sqrt[3]{7x+1}

Lösung

\int (1372 x) 3 7 x + 1 d x

Lösung

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

Schritte zur Lösung

\int 1372 x 3 7 x + 1 d x

Entferne die Konstante:

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

= 1372 \cdot \int x 3 7 x + 1 d x

Wende U-Substitution an

= 1372 \cdot \int \frac{3 u ( u - 1 )}{49} d u

Entferne die Konstante:

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

= 1372 \cdot \frac{1}{49} \cdot \int 3 u (u - 1) d u

Multipliziere aus

3 u (u - 1) : u^{\frac{4}{3}} - 3 u

= 1372 \cdot \frac{1}{49} \cdot \int u^{\frac{4}{3}} - 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

= 1372 \cdot \frac{1}{49} (\int u^{\frac{4}{3}} d u - \int 3 u d u)

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

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

= 1372 \cdot \frac{1}{49} (\frac{3}{7} u^{\frac{7}{3}} - \frac{3}{4} u^{\frac{4}{3}})

Setze in

u = 7 x + 1

ein

= 1372 \cdot \frac{1}{49} (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}})

Vereinfache

1372 \cdot \frac{1}{49} (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}}) : 28 (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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