19.6.1.12 Lebesgue measure and integration
Problem:
Calculate the Lebesgue integral \( \int_{[0,1]} f(x) d \mu \) from the function
\[
f(x)=\left\{\begin{array}{c}
x \cdot e^{x}, \quad x \in[0,1] \backslash Q \\
\tan ^{-1} \frac{1}{x+1}, \quad x \in Q \cap[0,1]
\end{array}\right.
\]