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^2-4x+9))/(x^2-6x+9)

Lösung

\int \frac{( x ^{2} - 4 x + 9 )}{x ^{2} - 6 x + 9} d x

Lösung

x + 2 ln ∣ x - 3 ∣ - \frac{6}{x - 3} + C

Schritte zur Lösung

\int \frac{x ^{2} - 4 x + 9}{x ^{2} - 6 x + 9} d x

Multipliziere aus

\frac{x ^{2} - 4 x + 9}{x ^{2} - 6 x + 9} : \frac{x ^{2}}{x ^{2} - 6 x + 9} - \frac{4 x}{x ^{2} - 6 x + 9} + \frac{9}{x ^{2} - 6 x + 9}

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{x ^{2}}{x ^{2} - 6 x + 9} d x - \int \frac{4 x}{x ^{2} - 6 x + 9} d x + \int \frac{9}{x ^{2} - 6 x + 9} d x

\int \frac{x ^{2}}{x ^{2} - 6 x + 9} d x = 6 ln ∣ x - 3 ∣ - \frac{9}{x - 3} + x

\int \frac{4 x}{x ^{2} - 6 x + 9} d x = 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3})

\int \frac{9}{x ^{2} - 6 x + 9} d x = - \frac{9}{x - 3}

= 6 ln ∣ x - 3 ∣ - \frac{9}{x - 3} + x - 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - \frac{9}{x - 3}

Vereinfache

6 ln ∣ x - 3 ∣ - \frac{9}{x - 3} + x - 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - \frac{9}{x - 3} : x + 2 ln ∣ x - 3 ∣ - \frac{6}{x - 3}

= x + 2 ln ∣ x - 3 ∣ - \frac{6}{x - 3}

Füge eine Konstante zur Lösung hinzu

= x + 2 ln ∣ x - 3 ∣ - \frac{6}{x - 3} + C

Graph

Beliebte Beispiele

inverselaplace ((3s+7))/(s^2+2s+4)

in v erse l a pl a ce \frac{( 3 s + 7 )}{s ^{2} + 2 s + 4}

derivative f(x)=-3x^2

d er i v a t i v e f (x) = - 3 x^{2}

tangent x^2+x,\at x=3

t an g e n t x^{2} + x, at x = 3

integral of 1/(x^2+6x+18)

\int \frac{1}{x ^{2} + 6 x + 18} d x

derivative of 7/2 x^2-5x+1

\frac{d}{d x} (\frac{7}{2} x^{2} - 5 x + 1)