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Beliebt Rechnen >

integral of x^2sqrt(5x+4)

Lösung

\int x^{2} 5 x + 4 d x

Lösung

\frac{1}{125} (\frac{2}{7} (5 x + 4)^{\frac{7}{2}} - \frac{16}{5} (5 x + 4)^{\frac{5}{2}} + \frac{32}{3} (5 x + 4)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x^{2} 5 x + 4 d x

Wende U-Substitution an

= \int \frac{u ( u - 4 ) ^{2}}{125} d u

Entferne die Konstante:

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

= \frac{1}{125} \cdot \int u (u - 4)^{2} d u

Multipliziere aus

u (u - 4)^{2} : u^{\frac{5}{2}} - 8 u^{\frac{3}{2}} + 16 u

= \frac{1}{125} \cdot \int u^{\frac{5}{2}} - 8 u^{\frac{3}{2}} + 16 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}{125} (\int u^{\frac{5}{2}} d u - \int 8 u^{\frac{3}{2}} d u + \int 16 u d u)

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

\int 8 u^{\frac{3}{2}} d u = \frac{16}{5} u^{\frac{5}{2}}

\int 16 u d u = \frac{32}{3} u^{\frac{3}{2}}

= \frac{1}{125} (\frac{2}{7} u^{\frac{7}{2}} - \frac{16}{5} u^{\frac{5}{2}} + \frac{32}{3} u^{\frac{3}{2}})

Setze in

u = 5 x + 4

ein

= \frac{1}{125} (\frac{2}{7} (5 x + 4)^{\frac{7}{2}} - \frac{16}{5} (5 x + 4)^{\frac{5}{2}} + \frac{32}{3} (5 x + 4)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{125} (\frac{2}{7} (5 x + 4)^{\frac{7}{2}} - \frac{16}{5} (5 x + 4)^{\frac{5}{2}} + \frac{32}{3} (5 x + 4)^{\frac{3}{2}}) + C

Graph

Beliebte Beispiele

fläche f(x)=x^3-6x^2+8,g(x)=x^2-4x

a re a f (x) = x^{3} - 6 x^{2} + 8, g (x) = x^{2} - 4 x

(dy)/(dx)=(x-8)e^{-2y}

\frac{d y}{d x} = (x - 8) e^{- 2 y}

derivative f(x)=2x+6

d er i v a t i v e f (x) = 2 x + 6

implicit 2x^2+xy+y^2=4

im pl i c i t 2 x^{2} + x y + y^{2} = 4

derivative f(x)=(x^2-4x+3)/(x+1)

d er i v a t i v e f (x) = \frac{x ^{2} - 4 x + 3}{x + 1}