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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Add a constant to the solution

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

\int \frac{1}{x ^{4} + y ^{2}} d y

\int 7 e^{- 3 x} d x

\frac{\partial}{\partial x} (arcsin (7 x yz))

x \to - 5 lim (\frac{3}{x - 5})

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

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