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Beliebt Rechnen >

integral of x^2sqrt(2x+3)

Lösung

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

Lösung

\frac{1}{35} (5 x^{2} - 6 x + 6) (2 x + 3)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u ( u - 3 ) ^{2}}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int u (u - 3)^{2} d u

Multipliziere aus

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

= \frac{1}{8} \cdot \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

= \frac{1}{8} (\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{1}{8} (\frac{2}{7} u^{\frac{7}{2}} - \frac{12}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}})

Setze in

u = 2 x + 3

ein

= \frac{1}{8} (\frac{2}{7} (2 x + 3)^{\frac{7}{2}} - \frac{12}{5} (2 x + 3)^{\frac{5}{2}} + 6 (2 x + 3)^{\frac{3}{2}})

Vereinfache

\frac{1}{8} (\frac{2}{7} (2 x + 3)^{\frac{7}{2}} - \frac{12}{5} (2 x + 3)^{\frac{5}{2}} + 6 (2 x + 3)^{\frac{3}{2}}) : \frac{1}{35} (5 x^{2} - 6 x + 6) (2 x + 3)^{\frac{3}{2}}

= \frac{1}{35} (5 x^{2} - 6 x + 6) (2 x + 3)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{35} (5 x^{2} - 6 x + 6) (2 x + 3)^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

derivative f(x)=3arccos(x/2)

d er i v a t i v e f (x) = 3 arccos (\frac{x}{2})

derivative of (3x/(5x^2-2x-9))

\frac{d}{d x} (\frac{3 x}{5 x ^{2} - 2 x - 9})

fläche y=x^2-18,y=17x

a re a y = x^{2} - 18, y = 17 x

integral from e to e^8 of 1/(xln(x^7))

\int_{e}^{e^{8}} \frac{1}{x ln ( x ^{7} )} d x

(dy)/(dx)=e^{-x}+y

\frac{d y}{d x} = e^{- x} + y