What is the integral of (5x^2-10x-8)/(x^3-4x) ?

The integral of (5x^2-10x-8)/(x^3-4x) is

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integral of (5x^2-10x-8)/(x^3-4x)

\int \frac{5 x ^{2} - 10 x - 8}{x ^{3} - 4 x} d x

\frac{5}{2} ln (∣ x + 2 ∣ ∣ x - 2 ∣) + \frac{5}{2} ln \frac{x}{2} + 1 - \frac{5}{2} ln \frac{x}{2} - 1 + 2 ln ∣ x ∣ - ln ∣ x + 2 ∣ - ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{5 x ^{2} - 10 x - 8}{x ^{3} - 4 x} d x

Expand

\frac{5 x ^{2} - 10 x - 8}{x ^{3} - 4 x} : \frac{5 x ^{2}}{x ^{3} - 4 x} - \frac{10 x}{x ^{3} - 4 x} - \frac{8}{x ^{3} - 4 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{5 x ^{2}}{x ^{3} - 4 x} d x - \int \frac{10 x}{x ^{3} - 4 x} d x - \int \frac{8}{x ^{3} - 4 x} d x

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

\int \frac{10 x}{x ^{3} - 4 x} d x = - 5 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1)

\int \frac{8}{x ^{3} - 4 x} d x = - 2 ln ∣ x ∣ + ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣

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

Simplify

\frac{5}{2} ln (∣ x + 2 ∣ ∣ x - 2 ∣) - (- 5 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1)) - (- 2 ln ∣ x ∣ + ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣) : \frac{5}{2} ln (∣ x + 2 ∣ ∣ x - 2 ∣) + \frac{5}{2} ln \frac{x}{2} + 1 - \frac{5}{2} ln \frac{x}{2} - 1 + 2 ln ∣ x ∣ - ln ∣ x + 2 ∣ - ln ∣ x - 2 ∣

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

Add a constant to the solution

= \frac{5}{2} ln (∣ x + 2 ∣ ∣ x - 2 ∣) + \frac{5}{2} ln \frac{x}{2} + 1 - \frac{5}{2} ln \frac{x}{2} - 1 + 2 ln ∣ x ∣ - ln ∣ x + 2 ∣ - ln ∣ x - 2 ∣ + C

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

(x^{2} + 1) \frac{d y}{d x} + 3 x (y - 1) = 0, y (0) = 4

\frac{\partial}{\partial x} (x cos (y ln (x)))

x \to 0.1 lim (\frac{1}{x})

\int \frac{1}{( 81 - x ^{2} ) ^{\frac{3}{2}}} d x

What is the integral of (5x^2-10x-8)/(x^3-4x) ?
The integral of (5x^2-10x-8)/(x^3-4x) is