Untegrate the following \( \frac{\int(x-1)^{2}}{\left(x^{2}+1\right)^{2}} d x \)

To solve the integral \( \int \frac{(x-1)^{2}}{(x^{2}+1)^{2}} \, dx \), we can use substitution and integration techniques. First, we expand the numerator: \[ (x-1)^2 = x^2 - 2x + 1 \] Thus, our integral becomes: \[ \int \frac{x^2 - 2x + 1}{(x^2 + 1)^2} \, dx \] This can be split into three separate integrals: \[ \int \frac{x^2}{(x^2 + 1)^2} \, dx - 2 \int \frac{x}{(x^2 + 1)^2} \, dx + \int \frac{1}{(x^2 + 1)^2} \, dx \] Now we evaluate these integrals one by one: 1. For \( \int \frac{x^2}{(x^2 + 1)^2} \, dx \): We can use the substitution \( u = x^2 + 1 \), which gives \( du = 2x \, dx\) or \( dx = \frac{du}{2x} \). 2. For \( -2 \int \frac{x}{(x^2 + 1)^2} \, dx \): This integral can be solved with the same substitution \( u = x^2 + 1 \). 3. For \( \int \frac{1}{(x^2 + 1)^2} \, dx \): This one can be solved using the formula: \[ \int \frac{1}{(x^2 + a^2)^2} \, dx = \frac{x}{2a^2(x^2 + a^2)} + C \] with \( a = 1 \). Finally, combining all these results will give the final answer for the original integral. Each integral yields specific parts of the function, creating a comprehensive and elegant final solution. In conclusion, the integral requires careful breakdown and appropriate substitutions, but it eventually leads to a manageable solution through systematic approaches.