Improper Integrals — Question 3

Question 3

Determine whether the integral converges. If it does, evaluate it. $$\int_{-1}^{1}\frac{dx}{\sqrt{|x|}}$$

Question 3 – Solution

Step 1: Split at the interior singularity. $$I=\int_{-1}^0\frac{dx}{\sqrt{|x|}}+\int_0^1\frac{dx}{\sqrt{|x|}}.$$ The integrand is even, so the two pieces are equal: $$I=2\int_0^1x^{-1/2}\,dx.$$ Step 2: Write the improper limit. $$I=2\lim_{\varepsilon\to0^+}\int_\varepsilon^1x^{-1/2}\,dx.$$ Step 3: Integrate and evaluate. $$\begin{align*} I&=2\lim_{\varepsilon\to0^+}[2x^{1/2}]_\varepsilon^1\\ &=4\lim_{\varepsilon\to0^+}(1-\sqrt{\varepsilon})=4. \end{align*}$$ $$\boxed{I=4\text{ (convergent)}}$$