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

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

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

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

- ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{- x - 3}{x ^{3} - x} : - \frac{x}{x ^{3} - x} - \frac{3}{x ^{3} - 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}{x ^{3} - x} d x - \int \frac{3}{x ^{3} - x} d x

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

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

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

Simplify

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

= - ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + 3 ln ∣ x ∣

Add a constant to the solution

= - ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ + C

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\int \frac{cos ( x )}{4 - sin ( x )} d x

\int_{1}^{4} x f (x) d x

\int \frac{x ^{2} + 4}{x ^{2}} d x

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