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+5x+4))/(x^2-2x+1)

Lösung

\int \frac{( x ^{2} + 5 x + 4 )}{x ^{2} - 2 x + 1} d x

Lösung

\frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} + 5 x + 4}{x ^{2} - 2 x + 1} : \frac{x ^{2}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + \frac{4}{x ^{2} - 2 x + 1}

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

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

\int \frac{5 x}{x ^{2} - 2 x + 1} d x = 5 (ln ∣ x - 1 ∣ - \frac{1}{x - 1})

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

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

Vereinfache

\frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + 5 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - \frac{4}{x - 1} : \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + C

Graph

Beliebte Beispiele

fläche y=x,y=5x-x^2

a re a y = x, y = 5 x - x^{2}

derivative of-2+2(-1^x)

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

derivative (-2nx^{n-1})/((1+x^n)^2)

d er i v a t i v e \frac{- 2 n x ^{n - 1}}{( 1 + x ^{n} ) ^{2}}

derivative of (dy/(dx))=y

\frac{d}{d x} (\frac{d y}{d x}) = y

integral of e^{-15x}

\int e^{- 15 x} d x