de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of x*sqrt(2x+1)

Lösung

\int x \cdot 2 x + 1 d x

Lösung

\frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x 2 x + 1 d x

Wende U-Substitution an

= \int \frac{u ( u - 1 )}{4} d u

Entferne die Konstante:

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

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

Multipliziere aus

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

= \frac{1}{4} \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}{4} (\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}{4} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 2 x + 1

ein

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

Vereinfache

\frac{1}{4} (\frac{2}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} (2 x + 1)^{\frac{3}{2}}) : \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

= \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

limit as x approaches 0 of (3(3-x)-9)/x

x \to 0 lim (\frac{3 ( 3 - x ) - 9}{x})

derivative f(x)=x^{9/2}e^x

d er i v a t i v e f (x) = x^{\frac{9}{2}} e^{x}

steigung (-2,-6),(0,6)

s l o p e (- 2, - 6), (0, 6)

integral of (6sqrt(x))

\int (6 x) d x

limit as x approaches 0 of (1+1/(2x))^x

x \to 0 lim ((1 + \frac{1}{2 x})^{x})