Improper Integrals Math Example 2

Solution

1. 1
   The integrand $x^{-1/2}$ is unbounded at $x = 0$, so replace the lower limit with $a > 0$ and take the limit.
2. 2
   Integrate $x^{-1/2}$: the antiderivative is $2\sqrt{x}$. Evaluate the definite integral: $$\lim_{a\to0^+}\left[2\sqrt{x}\right]_a^1 = \lim_{a\to0^+}(2\sqrt{1} - 2\sqrt{a})$$
3. 3
   Take the limit: $2\sqrt{a} \to 0$ as $a \to 0^+$, so the integral converges to $2 - 0 = 2$.