How to integrate x^2/(x^2-1)?

To integrate x^2/(x^2-1), use partial fractions to split the fraction into simpler terms.

First, factorise the denominator: x^2-1 = (x+1)(x-1)

Then, write the fraction as a sum of two fractions with denominators (x+1) and (x-1):

x^2/(x^2-1) = A/(x+1) + B/(x-1)

To find A and B, multiply both sides by (x+1)(x-1) and simplify:

x^2 = A(x-1) + B(x+1)

Let x = 1: 1 = 2B, so B = 1/2
Let x = -1: 1 = -2A, so A = -1/2

Therefore, x^2/(x^2-1) = -1/2/(x+1) + 1/2/(x-1)

Now, integrate each term separately:

∫-1/2/(x+1) dx = -1/2 ln|x+1| + C1
∫1/2/(x-1) dx = 1/2 ln|x-1| + C2

Where C1 and C2 are constants of integration.

Therefore, the final answer is:

∫x^2/(x^2-1) dx = -1/2 ln|x+1| + 1/2 ln|x-1| + C