de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (2x^2+x+1)/((x+3)(x-1)^2)

Lösung

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

Lösung

ln ∣ x + 3 ∣ + ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

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

= ln ∣ x + 3 ∣ + ln ∣ x - 1 ∣ - \frac{1}{x - 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 3 ∣ + ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Graph

Beliebte Beispiele

sum from n=1 to infinity of 16^nx^nn!

n = 1 \sum \infty 1 6^{n} x^{n} n!

integral of xsqrt(x^2+7)

\int x x^{2} + 7 d x

integral of 1/(2000(2000-x))

\int \frac{1}{2000 ( 2000 - x )} d x

derivative f(x)=x^a(1-x)^b

d er i v a t i v e f (x) = x^{a} (1 - x)^{b}

tangent f(x)=9-4x+6x^2,\at x=8

t an g e n t f (x) = 9 - 4 x + 6 x^{2}, at x = 8