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Beliebt Rechnen >

integral of 3xsqrt(x+5)

Lösung

\int 3 x x + 5 d x

Lösung

\frac{6}{5} (x + 5)^{\frac{5}{2}} - 10 (x + 5)^{\frac{3}{2}} + C

Schritte zur Lösung

\int 3 x x + 5 d x

Entferne die Konstante:

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

= 3 \cdot \int x x + 5 d x

Wende U-Substitution an

= 3 \cdot \int (u - 5) u d u

Multipliziere aus

(u - 5) u : u^{\frac{3}{2}} - 5 u

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

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

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= 3 (\frac{2}{5} u^{\frac{5}{2}} - \frac{10}{3} u^{\frac{3}{2}})

Setze in

u = x + 5

ein

= 3 (\frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}})

Vereinfache

3 (\frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}}) : \frac{6}{5} (x + 5)^{\frac{5}{2}} - 10 (x + 5)^{\frac{3}{2}}

= \frac{6}{5} (x + 5)^{\frac{5}{2}} - 10 (x + 5)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} (x + 5)^{\frac{5}{2}} - 10 (x + 5)^{\frac{3}{2}} + C

Graph

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