Lời giải

\( I = \int_0^3 \sqrt{x(x^2 - 2x + 1)} \, dx = \int_0^3 \sqrt{x(x - 1)^2} \, dx \)

\(= \int_0^3 |x - 1| \sqrt{x}\, dx\)

\(= \int_0^1 (1 - x) \sqrt{x} \, dx + \int_1^3 (x - 1) \sqrt{x} \, dx\)

\(= \int_0^1 (x^{\frac{1}{2}} - x^{\frac{3}{2}}) \, dx + \int_1^3 (x^{\frac{3}{2}} - x^{\frac{1}{2}}) \, dx\)

\(= \left( \frac{2}{3} x \sqrt{x}- \frac{2}{5} x^2 \sqrt{x}\right)\bigg|_0^1 + \left( \frac{2}{5} x^2 \sqrt{x} - \frac{2}{3} x \sqrt{x} \right)\bigg|_1^3\)

\(= \frac{8 + 24\sqrt{3}}{15}\)

* Bấm 
•  \( \int_0^3 \sqrt{x^3 - 2x^2 + x} \, dx = 3.304614\)

•  \( \frac{8 + 24.3^{\frac{1}{2}}}{15} = 3.304614 \)