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

Lösung

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

Lösung

\frac{6}{5} ln ∣ x - 3 ∣ - \frac{1}{10} ln x^{2} + 1 + \frac{2}{5} arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x ^{2}}{( x - 3 ) ( x ^{2} + 1 )} d x = \frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x))

\int \frac{x}{( x - 3 ) ( x ^{2} + 1 )} d x = \frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x))

= \frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x)) + \frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x))

Vereinfache

\frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x)) + \frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x)) : \frac{6}{5} ln ∣ x - 3 ∣ - \frac{1}{10} ln x^{2} + 1 + \frac{2}{5} arctan (x)

= \frac{6}{5} ln ∣ x - 3 ∣ - \frac{1}{10} ln x^{2} + 1 + \frac{2}{5} arctan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} ln ∣ x - 3 ∣ - \frac{1}{10} ln x^{2} + 1 + \frac{2}{5} arctan (x) + C

Graph

Beliebte Beispiele

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

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

derivative of (x^2/(20))

\frac{d}{d x} (\frac{x ^{2}}{20})

derivative of 3sqrt(x)-4x

\frac{d}{d x} (3 x - 4 x)

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

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

steigung (-3/7 ,0),(0,6)

s l o p e (- \frac{3}{7}, 0), (0, 6)