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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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