English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

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

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

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende integrale Substitution an

= 2 \cdot \int \frac{- v + 3}{2 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{- v + 3}{v ^{2} + 1} d v

Multipliziere aus

\frac{- v + 3}{v ^{2} + 1} : - \frac{v}{v ^{2} + 1} + \frac{3}{v ^{2} + 1}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot \frac{1}{2} (- \int \frac{v}{v ^{2} + 1} d v + \int \frac{3}{v ^{2} + 1} d v)

\int \frac{v}{v ^{2} + 1} d v = \frac{1}{2} ln v^{2} + 1

\int \frac{3}{v ^{2} + 1} d v = 3 arctan (v)

= 2 \cdot \frac{1}{2} (- \frac{1}{2} ln v^{2} + 1 + 3 arctan (v))

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e 4 x^{3} - 5 7 x^{6} + π x

l a pl a ce t r an s f or m e^{- 4 t} cos (3 t)

f (t) = 6 cos (t)

\frac{d}{d x} (18 x (x^{2} + 6)^{8})

\int x^{1.5} d x