Question

Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

Hint:

For integrals involving square roots of trigonometric functions, consider substituting and using standard integrals for trigonometric powers.

Answer 1

The expression within the square root is first simplified: \[ \sqrt{2 \sin 2x} = \sqrt{4 \sin x \cos x} = 2 \sqrt{\sin x \cos x} \] The integral is then rewritten as: \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \cdot 2 \sqrt{\sin x \cos x}} = \frac{1}{2} \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^{\frac{5}{2}} x \sqrt{\sin x}} \] A substitution \( u = \sin x \), leading to \( du = \cos x \, dx \), is then employed. Adjusting the integration limits and performing the integration yields the final answer.