Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$

Question

How do we calculate the following integral:

$$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$

Answer 1

HINT: $$|x-a|=\begin{cases} x-a &\text{ if } x-a\ge 0 \text{ i.e., if } x\ge a\ \\ -(x-a) &\text{ if } x-a<0 \end{cases}$$

Answer 2

Hint:

$$\int_0^5\ldots=\int_0^1\ldots+\int_1^2\ldots+\int_2^4\ldots+\int_4^5\ldots$$