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Beliebt Rechnen >

integral of x^7sqrt(x^4+3)

Lösung

\int x^{7} x^{4} + 3 d x

Lösung

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

Schritte zur Lösung

\int x^{7} x^{4} + 3 d x

Wende U-Substitution an

= \int \frac{( u - 3 ) u}{4} d u

Entferne die Konstante:

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

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

Multipliziere aus

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

= \frac{1}{4} \cdot \int u^{\frac{3}{2}} - 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}{4} (\int u^{\frac{3}{2}} d u - \int 3 u d u)

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

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

= \frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} - 2 u^{\frac{3}{2}})

Setze in

u = x^{4} + 3

ein

= \frac{1}{4} (\frac{2}{5} (x^{4} + 3)^{\frac{5}{2}} - 2 (x^{4} + 3)^{\frac{3}{2}})

Vereinfache

\frac{1}{4} (\frac{2}{5} (x^{4} + 3)^{\frac{5}{2}} - 2 (x^{4} + 3)^{\frac{3}{2}}) : \frac{1}{10} (x^{4} - 2) (x^{4} + 3)^{\frac{3}{2}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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