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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int u (u - 2) d u

Multipliziere aus

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

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

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

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

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

Setze in

u = x + 3

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\int_{1}^{5} 1 + 2^{2} d x