The integral $$\int \left(\frac{x-1}{x+1}\right)^4\,dx$$ can be solved by substituting $$u=x+1$$, which simplifies the expression to $$\left(1-2u^{-1}\right)^4$$. Applying the binomial theorem results in the expansion $$16u^{-4}-32u^{-3}+24u^{-2}-8u^{-1}+1$$. The integration can then be performed term by term, followed by back-substitution for $$u$$ to obtain the final result.
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