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

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

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

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

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

Solution steps

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

Expand

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

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

\int \frac{8 x}{x ^{3} - 4 x} d x = - 4 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1)

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

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

Simplify

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

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

Add a constant to the solution

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

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

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\int \frac{d}{1} (cos^{3} (x)) d x

\int x^{3} 7^{x^{4}} d x

\int \frac{1}{2 x ^{2} + 9} d x

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