Ta có: \(\int\limits_1^m {\left( {{x^2} - 2x + 5} \right)dx}  = \left. {\left( {\frac{{{x^3}}}{3} - {x^2} + 5x} \right)} \right|_1^m = \frac{{{m^3} - 1}}{3} - \left( {{m^2} - 1} \right) + 5\left( {m - 1} \right) = \frac{{32}}{3} \Rightarrow m = 3.\)