How can I prove that $\int_0^1 \frac{1}{\sqrt{|x-1/2|}} \, dx < \infty.$ in Lebesgue measure?

Question

How can I prove that $\int_0^1 \frac{1}{\sqrt{|x-1/2|}} \, dx < \infty.$ in Lebesgue measure?

My thought:

Using the Monotone Convergence theorem, but still I am stuck in applying it, could anyone help me please?

Answer 1

$$\int_0^{1/2}\frac{1}{\sqrt{\frac{1}{2}-x}}\,\mathrm d x\quad \text{and}\quad \int_{1/2}^1\frac{1}{\sqrt{x-\frac{1}{2}}}\,\mathrm d x$$ are convergent improper integrals of positives functions. So, obviously $x\mapsto \frac{1}{\sqrt{|\frac{1}{2}-x|}}$ is Lebesgue integrable on $(0,1)$.