9.2. \( \int \frac{x^{2}-1}{x(x-1)^{2}} d x \)

Calculate the integral \( \int \frac{x^{2}-1}{x(x-1)^{2}} dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{x^{2}-1}{x\left(x-1\right)^{2}} dx\) - step1: Rewrite the fraction: \(\int \left(-\frac{1}{x}+\frac{2}{x-1}\right) dx\) - step2: Use properties of integrals: \(\int -\frac{1}{x} dx+\int \frac{2}{x-1} dx\) - step3: Evaluate the integral: \(-\ln{\left(\left|x\right|\right)}+\int \frac{2}{x-1} dx\) - step4: Evaluate the integral: \(-\ln{\left(\left|x\right|\right)}+2\ln{\left(\left|x-1\right|\right)}\) - step5: Use the logarithm base change rule: \(-\ln{\left(\left|x\right|\right)}+\ln{\left(\left(x-1\right)^{2}\right)}\) - step6: Reorder the terms: \(\ln{\left(\left(x-1\right)^{2}\right)}-\ln{\left(\left|x\right|\right)}\) - step7: Transform the expression: \(\ln{\left(\frac{\left(x-1\right)^{2}}{\left|x\right|}\right)}\) - step8: Add the constant of integral C: \(\ln{\left(\frac{\left(x-1\right)^{2}}{\left|x\right|}\right)} + C, C \in \mathbb{R}\) The integral of \( \frac{x^{2}-1}{x(x-1)^{2}} \) with respect to \( x \) is \( \ln\left(\frac{(x-1)^{2}}{|x|}\right) + C \), where \( C \) is an arbitrary constant.