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

Lösung

\int 3 y^{5} y^{3} + 1 d y

\frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int 3 y^{5} y^{3} + 1 d y

Entferne die Konstante:

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

= 3 \cdot \int y^{5} y^{3} + 1 d y

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

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

= 3 \cdot \frac{1}{3} \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

= 3 \cdot \frac{1}{3} (\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}}

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

Setze in

u = y^{3} + 1

ein

= 3 \cdot \frac{1}{3} (\frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}})

Vereinfache

3 \cdot \frac{1}{3} (\frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}}) : \frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}}

= \frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (y^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (y^{3} + 1)^{\frac{3}{2}} + C

x \to - 1 lim (x + 2 x^{4})

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