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Beliebt Rechnen >

integral of xsqrt(4x+1)

Lösung

\int x 4 x + 1 d x

Lösung

\frac{1}{60} (6 x - 1) (4 x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x 4 x + 1 d x

Wende U-Substitution an

= \int \frac{u ( u - 1 )}{16} d u

Entferne die Konstante:

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

= \frac{1}{16} \cdot \int u (u - 1) d u

Multipliziere aus

u (u - 1) : u^{\frac{3}{2}} - u

= \frac{1}{16} \cdot \int u^{\frac{3}{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

= \frac{1}{16} (\int u^{\frac{3}{2}} d u - \int u d u)

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

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

= \frac{1}{16} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 4 x + 1

ein

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

Vereinfache

\frac{1}{16} (\frac{2}{5} (4 x + 1)^{\frac{5}{2}} - \frac{2}{3} (4 x + 1)^{\frac{3}{2}}) : \frac{1}{60} (6 x - 1) (4 x + 1)^{\frac{3}{2}}

= \frac{1}{60} (6 x - 1) (4 x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{60} (6 x - 1) (4 x + 1)^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

f(x)=sqrt(5)x+sqrt(7x)

f (x) = 5 x + 7 x

(x^2+1)y^'+3x(y-1)=0,y(0)=4

(x^{2} + 1) y^{^{'}} + 3 x (y - 1) = 0, y (0) = 4

integral of (x^3+1)/(x^3+x^2)

\int \frac{x ^{3} + 1}{x ^{3} + x ^{2}} d x

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d er i v a t i v e y = e^{- x} arctan (5 e^{x})

derivative x^2ln(x^2+1)

d er i v a t i v e x^{2} ln (x^{2} + 1)