What is the integral of (x^3-3x-1)/(x^3+x^2-2x) ?

The integral of (x^3-3x-1)/(x^3+x^2-2x) is x-3/2 ln|x+2|+1/2 ln|x|+ln| 2/3 x+4/3 |-ln| 2/3 x-2/3 |+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

x - \frac{3}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + ln \frac{2}{3} x + \frac{4}{3} - ln \frac{2}{3} x - \frac{2}{3} + C

Solution steps

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

Expand

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

Apply the Sum Rule:

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

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

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

\int \frac{3 x}{x ^{3} + x ^{2} - 2 x} d x = - ln \frac{4}{3} + \frac{2}{3} x + ln - \frac{2}{3} + \frac{2}{3} x

\int \frac{1}{x ^{3} + x ^{2} - 2 x} d x = - \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣

= x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{4}{3} ln ∣ x + 2 ∣ - (- ln \frac{4}{3} + \frac{2}{3} x + ln - \frac{2}{3} + \frac{2}{3} x) - (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣)

Simplify

x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{4}{3} ln ∣ x + 2 ∣ - (- ln \frac{4}{3} + \frac{2}{3} x + ln - \frac{2}{3} + \frac{2}{3} x) - (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣) : x - \frac{3}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + ln \frac{2}{3} x + \frac{4}{3} - ln \frac{2}{3} x - \frac{2}{3}

= x - \frac{3}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + ln \frac{2}{3} x + \frac{4}{3} - ln \frac{2}{3} x - \frac{2}{3}

Add a constant to the solution

= x - \frac{3}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + ln \frac{2}{3} x + \frac{4}{3} - ln \frac{2}{3} x - \frac{2}{3} + C

\int e^{3} + e^{x} d x

\int \frac{( ln ( 3 x ) ) ^{2}}{x} d x

\int sin (\frac{y}{x}) d y

\int x e^{u} d x

\int (4 sinh (x) + x^{7}) d x

What is the integral of (x^3-3x-1)/(x^3+x^2-2x) ?
The integral of (x^3-3x-1)/(x^3+x^2-2x) is x-3/2 ln|x+2|+1/2 ln|x|+ln| 2/3 x+4/3 |-ln| 2/3 x-2/3 |+C