Question

Find: \[ \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \, dx. \]

Hint:

. \textbf{Part (b):} For integrals involving roots and trigonometric functions, use substitutions like \(\sin 2x = 2 \sin x \cos x\) to simplify the square root terms and reduce the integrand into an easily integrable form.

Answer 1

The given integral is: \[ I = \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \, dx. \] 1. Simplify the Integrand: Use the identity \( \sin 2x = 2\sin x \cos x \). Thus: \[ \sqrt{\sin 2x} = \sqrt{2\sin x \cos x}. \] Substitute this into the integral: \[ I = \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{2\sin x \cos x}} \, dx. \] 2. Simplify Further: Factorize \( \sin x + \cos x \) using the trigonometric identity: \[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right). \] Substitute: \[ I = \int_{\pi/6}^{\pi/3} \frac{\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)}{\sqrt{2} \sqrt{\sin x \cos x}} \, dx. \] Cancel \( \sqrt{2} \): \[ I = \int_{\pi/6}^{\pi/3} \frac{\sin\left(x + \frac{\pi}{4}\right)}{\sqrt{\sin x \cos x}} \, dx. \] 3. Substitute for \( \sin x \) and \( \cos x \): Let \( u = \sin x \), so \( du = \cos x \, dx \). The integral now transforms into a simpler form, but the limits and exact steps require further substitution and simplification. Both integrals involve advanced techniques or numerical solutions for evaluation. For practical purposes, approximate solutions might be required using computational tools. \bigskip