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Beliebt Rechnen >

integral of (3x)^5sqrt((3x)^3+1)

Lösung

\int (3 x)^{5} (3 x)^{3} + 1 d x

Lösung

\frac{1}{9} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int (3 x)^{5} (3 x)^{3} + 1 d x

Vereinfache

(3 x)^{5} (3 x)^{3} + 1 : 243 x^{5} 27 x^{3} + 1

= \int 243 x^{5} 27 x^{3} + 1 d x

Entferne die Konstante:

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

= 243 \cdot \int x^{5} 27 x^{3} + 1 d x

Wende U-Substitution an

= 243 \cdot \int \frac{u ( u - 1 )}{2187} d u

Entferne die Konstante:

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

= 243 \cdot \frac{1}{2187} \cdot \int u (u - 1) d u

Multipliziere aus

u (u - 1) : u^{\frac{3}{2}} - u

= 243 \cdot \frac{1}{2187} \cdot \int u^{\frac{3}{2}} - 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

= 243 \cdot \frac{1}{2187} (\int u^{\frac{3}{2}} d u - \int u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

= 243 \cdot \frac{1}{2187} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 27 x^{3} + 1

ein

= 243 \cdot \frac{1}{2187} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}})

Vereinfache

243 \cdot \frac{1}{2187} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}}) : \frac{1}{9} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}})

= \frac{1}{9} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (\frac{2}{5} (27 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (27 x^{3} + 1)^{\frac{3}{2}}) + C

Graph

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