¡ìdx/(x^2-1)^2
(1/4)[-2x/(x^2-1) - ln(1-x)+ln(1+x) ] + c
used Wolfram alpha
And you arrive there by using partial fractions:
1/(x^2-1)^2 = 1/4 (1/(x+1) + 1/(x+1)^2 - 1/(x-1) + 1/(x-1)^2)
To integrate the expression ∫(dx)/(x^2-1)^2, we can use a technique called partial fraction decomposition.
Here's how to approach it:
Step 1: Factorize the denominator
The first step is to factorize the expression in the denominator, (x^2-1)^2. We can observe that (x^2-1)^2 is a perfect square. Using the difference of squares formula, we can express it as (x-1)^2 * (x+1)^2.
Step 2: Write the expression as a sum of partial fractions
Once we have factored the denominator, we can write the given expression as the sum of partial fractions. We do this by assuming that the given expression can be expressed as A/(x-1) + B/(x+1) + C/(x-1)^2 + D/(x+1)^2, where A, B, C, and D are constants that we need to determine.
Step 3: Simplify the expression and find the numerator constants
To find the constant values A, B, C, and D, we can multiply both sides of the equation by the least common denominator, (x-1)^2 * (x+1)^2. This will help us eliminate the denominators and solve for the constants.
Step 4: Integration of the partial fractions
Once we have determined the constant values A, B, C, and D, we can integrate each term individually. The integral of A/(x-1) will be A * ln|x-1|, the integral of B/(x+1) will be B * ln|x+1|, the integral of C/(x-1)^2 will be -C/(x-1), and the integral of D/(x+1)^2 will be -D/(x+1), based on the general integration rules.
Step 5: Combine the integrals
Finally, combine all the integrals obtained from each partial fraction to find the complete integral of the original expression.
It's important to note that the actual integration process may involve slightly different steps depending on the specific form of the expression and the chosen method. However, the steps mentioned above outline the general approach for solving the given integral.