19.6.1.14 Lebesgue measure and integration

Condition: Find the Lebesgue integral \( \int_{[0 ; 2]} f d \mu \quad \) of the function \( f(x)=\left\{\begin{aligned} x^{5}, & \text { if } x \text { rational, } \\ x, & \text { if } x \text { irrational algebraic, } \\ x^{2}, & \text { if } x \text { transcendental. }\end{aligned}\right. \)