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(2-x-x^2))

Lösung

\int \frac{x}{2 - x - x ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{x}{2 - x - x ^{2}} d x

Vervollständige das Quadrat

2 - x - x^{2} : - (x + \frac{1}{2})^{2} + \frac{9}{4}

= \int \frac{x}{- ( x + \frac{1}{2} ) ^{2} + \frac{9}{4}} d x

Wende U-Substitution an

= \int \frac{2 u - 1}{- 4 u ^{2} + 9} d u

Multipliziere aus

\frac{2 u - 1}{- 4 u ^{2} + 9} : \frac{2 u}{- 4 u ^{2} + 9} - \frac{1}{- 4 u ^{2} + 9}

= \int \frac{2 u}{- 4 u ^{2} + 9} - \frac{1}{- 4 u ^{2} + 9} 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 \frac{2 u}{- 4 u ^{2} + 9} d u - \int \frac{1}{- 4 u ^{2} + 9} d u

\int \frac{2 u}{- 4 u ^{2} + 9} d u = - \frac{1}{2} - 4 u^{2} + 9

\int \frac{1}{- 4 u ^{2} + 9} d u = \frac{1}{2} arcsin (\frac{2}{3} u)

= - \frac{1}{2} - 4 u^{2} + 9 - \frac{1}{2} arcsin (\frac{2}{3} u)

Setze in

u = x + \frac{1}{2}

ein

= - \frac{1}{2} - 4 (x + \frac{1}{2})^{2} + 9 - \frac{1}{2} arcsin (\frac{2}{3} (x + \frac{1}{2}))

Vereinfache

- \frac{1}{2} - 4 (x + \frac{1}{2})^{2} + 9 - \frac{1}{2} arcsin (\frac{2}{3} (x + \frac{1}{2})) : - - x^{2} - x + 2 - \frac{1}{2} arcsin (\frac{2}{3} x + \frac{1}{3})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

xy^2(dy)/(dx)=y^3+x^3

x y^{2} \frac{d y}{d x} = y^{3} + x^{3}

inverselaplace 1/(s(s^2+5))

in v erse l a pl a ce \frac{1}{s ( s ^{2} + 5 )}

limit as x approaches 0 of sin(3x)

x \to 0 lim (sin (3 x))

limit as j approaches infinity of (2e^j+e^{-j}+2e^4+1)/(e^j+e^4)

j \to \infty lim (\frac{2 e ^{j} + e ^{- j} + 2 e ^{4} + 1}{e ^{j} + e ^{4}})

derivative y=sec^2(sqrt(x))

d er i v a t i v e y = sec^{2} (x)