15 Multiple Choice 1 point Determine if the integral diverges or converges. Evaluate the integral if it converges. \( \int_{1}^{\infty} \frac{3}{\sqrt{x}} d x \) \( 2 / 5 \) 1 This integral diverges. 0

To determine whether the integral \( \int_{1}^{\infty} \frac{3}{\sqrt{x}} dx \) converges or diverges, we first compute the integral. When we evaluate it, we find: \[ \int \frac{3}{\sqrt{x}} dx = 6\sqrt{x} + C \] Now, we evaluate the definite integral from 1 to \(\infty\): \[ \lim_{b \to \infty} \left( 6\sqrt{b} - 6\sqrt{1} \right) = \lim_{b \to \infty} (6\sqrt{b} - 6) = \infty \] Since the value approaches infinity, the integral diverges. Now, think of it like this: integrating \(\frac{3}{\sqrt{x}}\) is like trying to fill a tub with a hole in the bottom—it just keeps leaking (diverging) no matter how much water (area) you pour into it! Nevertheless, if you love exploring integrals, remember that improper integrals often lead to surprises. So, keep that adventurous spirit alive while diving into more complex functions!