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^2)/(sqrt(x^2-x+1))

Lösung

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

Lösung

\frac{1}{16} (8 x x^{2} - x + 1 + 12 x^{2} - x + 1 - 6 ln 2 x + 3 \frac{4 x ^{2} - 4 x + 4}{3} - 1 + 4 ln \frac{2 x}{3} + \frac{4 x ^{2} - 4 x + 4}{3} - \frac{1}{3} + 3 ln (3)) + C

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{( 2 u + 1 ) ^{2}}{4 u ^{2} + 3} : \frac{4 u ^{2}}{4 u ^{2} + 3} + \frac{4 u}{4 u ^{2} + 3} + \frac{1}{4 u ^{2} + 3}

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

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

\int \frac{4 u}{4 u ^{2} + 3} d u = 4 u^{2} + 3

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

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

Setze in

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

ein

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

Vereinfache

\frac{1}{2} \frac{1}{4} 2 (x - \frac{1}{2}) 4 (x - \frac{1}{2})^{2} + 3 - 3 ln 2 (x - \frac{1}{2}) + 3 \frac{4 ( x - \frac{1}{2} ) ^{2} + 3}{3} - \frac{ln ( 3 )}{2} + 4 (x - \frac{1}{2})^{2} + 3 + \frac{1}{2} ln \frac{2}{3} (x - \frac{1}{2}) + \frac{1}{3} (3 + 4 (x - \frac{1}{2})^{2}) : \frac{1}{16} (8 x x^{2} - x + 1 + 12 x^{2} - x + 1 - 6 ln 2 x + 3 \frac{4 x ^{2} - 4 x + 4}{3} - 1 + 4 ln \frac{2 x}{3} + \frac{4 x ^{2} - 4 x + 4}{3} - \frac{1}{3} + 3 ln (3))

= \frac{1}{16} (8 x x^{2} - x + 1 + 12 x^{2} - x + 1 - 6 ln 2 x + 3 \frac{4 x ^{2} - 4 x + 4}{3} - 1 + 4 ln \frac{2 x}{3} + \frac{4 x ^{2} - 4 x + 4}{3} - \frac{1}{3} + 3 ln (3))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{16} (8 x x^{2} - x + 1 + 12 x^{2} - x + 1 - 6 ln 2 x + 3 \frac{4 x ^{2} - 4 x + 4}{3} - 1 + 4 ln \frac{2 x}{3} + \frac{4 x ^{2} - 4 x + 4}{3} - \frac{1}{3} + 3 ln (3)) + C

Graph

Beliebte Beispiele

integral of x^4sqrt(9+x^5)

\int x^{4} 9 + x^{5} d x

integral of 11x^{4/7}+4x^{-6/7}

\int 11 x^{\frac{4}{7}} + 4 x^{- \frac{6}{7}} d x

derivative of (x^2/7)

\frac{d}{d x} (\frac{x ^{2}}{7})

tangent f(x)=2x^3-4x^2,\at x=1

t an g e n t f (x) = 2 x^{3} - 4 x^{2}, at x = 1

integral from 1 to 5 of 5/(x^2+4x+4)

\int_{1}^{5} \frac{5}{x ^{2} + 4 x + 4} d x