Thực hiện phép tính: j) 1 + (x^3 − x )/(x^2 + 1) ⋅ ( 1/( 1 − x) − 1/( 1 − x^2) ) .

j) \(1 + \frac{{{x^3} - x}}{{{x^2} + 1}} \cdot \left( {\frac{1}{{1 - x}} - \frac{1}{{1 - {x^2}}}} \right) = 1 + \frac{{x\left( {{x^2} - 1} \right)}}{{{x^2} + 1}} \cdot \left[ {\frac{1}{{1 - x}} - \frac{1}{{\left( {1 - x} \right)\left( {1 + x} \right)}}} \right]\)

\( = 1 - \frac{{x\left( {1 - {x^2}} \right)}}{{{x^2} + 1}} \cdot \frac{{1 + x - 1}}{{\left( {1 - x} \right)\left( {1 + x} \right)}} = 1 - \frac{{x\left( {1 - x} \right)\left( {1 + x} \right)}}{{{x^2} + 1}} \cdot \frac{x}{{\left( {1 - x} \right)\left( {1 + x} \right)}}\)

\( = 1 - \frac{{{x^2}}}{{{x^2} + 1}} = \frac{{{x^2} + 1 - {x^2}}}{{{x^2} + 1}} = \frac{1}{{{x^2} + 1}}.\)