What is the integral of ((4x-2))/(x^3-x^2-2x) ?

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

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integral of ((4x-2))/(x^3-x^2-2x)

\int \frac{( 4 x - 2 )}{x ^{3} - x ^{2} - 2 x} d x

ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{4 x - 2}{x ^{3} - x ^{2} - 2 x} : \frac{1}{x} - \frac{2}{x + 1} + \frac{1}{x - 2}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣

Add a constant to the solution

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣ + C

\frac{d}{d x} (\frac{5}{3} x^{3} + x^{2} - 1)

\int \frac{sin ^{7} ( ln ( x ) )}{x} d x

\frac{1}{x ^{8}} y^{^{'}} = 12 + 15 y

\frac{\partial}{\partial x} (x^{2} y + x y^{2})

y = x^{3}, at x = - 9

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