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Beliebt Rechnen >

integral of y^2sqrt(y+3)

Lösung

\int y^{2} y + 3 d y

Lösung

\frac{2}{7} (y + 3)^{\frac{7}{2}} - \frac{12}{5} (y + 3)^{\frac{5}{2}} + 6 (y + 3)^{\frac{3}{2}} + C

Schritte zur Lösung

\int y^{2} y + 3 d y

Wende U-Substitution an

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

Multipliziere aus

(u - 3)^{2} u : u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 u

= \int u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 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{5}{2}} d u - \int 6 u^{\frac{3}{2}} d u + \int 9 u d u

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 6 u^{\frac{3}{2}} d u = \frac{12}{5} u^{\frac{5}{2}}

\int 9 u d u = 6 u^{\frac{3}{2}}

= \frac{2}{7} u^{\frac{7}{2}} - \frac{12}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}}

Setze in

u = y + 3

ein

= \frac{2}{7} (y + 3)^{\frac{7}{2}} - \frac{12}{5} (y + 3)^{\frac{5}{2}} + 6 (y + 3)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (y + 3)^{\frac{7}{2}} - \frac{12}{5} (y + 3)^{\frac{5}{2}} + 6 (y + 3)^{\frac{3}{2}} + C

Graph

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