Step-by-Step Solution
Key Concept: Use substitution u = \sqrt{sin} x, then du = (1/2\sqrt{sin} x) * cos x dx. The integral simplifies to 2 \int e^u du.
Let I = \int\sqrt{cot} x e^\sqrt{sin} x \sqrt{cos} x dx. Since \sqrt{cot} x = \sqrt{cos} x / \sqrt{sin} x, we have I = \int (\sqrt{cos} x / \sqrt{sin} x) * e^\sqrt{sin} x * \sqrt{cos} x dx = \int (cos x / \sqrt{sin} x) * e^\sqrt{sin} x dx. Let u = \sqrt{sin} x, then du = (1 / 2\sqrt{sin} x) * cos x dx, so (cos x / \sqrt{sin} x) dx = 2 du. Thus, I = \int e^u * 2 du = 2e^u + C = 2e^\sqrt{sin} x + C.
Correct Answer: 2