# How do you integrate #int (x-1)/( x^4 (x-1)^2)# using partial fractions?

How about NOT using partial fractions and instead using the binomial theorem (or the sum of geometrical series, same thing) after cancelling the

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- I integrated using the fundamental fractions method.

I broke down the integrand into simpler fractions.

I distinguished between the two sides.

I distinguished between the two sides.

I distinguished between the two sides.

Thus,

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To integrate ( \frac{x-1}{x^4(x-1)^2} ) using partial fractions, we first express the given rational function as a sum of partial fractions.

( \frac{x-1}{x^4(x-1)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{E}{x-1} + \frac{F}{(x-1)^2} )

Next, we find the values of ( A ), ( B ), ( C ), ( D ), ( E ), and ( F ) by equating coefficients.

Once we have found the values of ( A ), ( B ), ( C ), ( D ), ( E ), and ( F ), we integrate each term separately.

After integrating each term, we combine the results to obtain the final integral of the given function.

However, it's important to note that due to the complexity of the partial fraction decomposition and integration process, the detailed steps for solving this integral may be lengthy and involve several algebraic manipulations.

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