Question

Evaluate the integral: \[ \int \frac{dx}{\cos x \sqrt{\cos 2x}}. \]

• A. \( \frac{1}{2} \log \left| \tan \left( \frac{\pi}{4} + x \right) \right| + c \)
• B. \( \frac{1}{2} \log \left| 1 - \tan x \right| + c \)
• C. \( 2 \log \left| 1 + \tan x \right| + c \)
• D. \( \sin^{-1} (\tan x) + c \)

Hint:

For integrals involving trigonometric functions, use trigonometric identities and substitutions to simplify the integrand.

Answer 1

The Correct Option is D

Step 1: Simplify the integrand.
We are asked to evaluate the integral: \[ I = \int \frac{dx}{\cos x \sqrt{\cos 2x}}. \] We can use the identity \( \cos 2x = 2\cos^2 x - 1 \) to express \( \cos 2x \) in terms of \( \cos x \). This simplifies the integrand and allows us to rewrite the expression in a more manageable form.
Step 2: Apply a substitution.
We then use the substitution \( u = \tan x \), so \( du = \sec^2 x \, dx \), to transform the integral into a standard form. After solving the integral, we get the result: \[ I = \sin^{-1} (\tan x) + c. \]
Step 3: Conclusion.
Thus, the solution to the integral is \( \sin^{-1} (\tan x) + c \), which corresponds to option (D).