Question

The integral \( \int \frac{\csc x}{\cos^2\left(1 + \log \tan \frac{x}{2}\right)} \, dx \) is equal to:

• A. \( \sin^2(1 + \log \tan \frac{x}{2}) + C \)
• B. \( \tan(1 + \log \tan \frac{x}{2}) + C \)
• C. \( -\tan(1 + \log \tan \frac{x}{2}) + C \)
• D. \( \sec^2(1 + \log \tan \frac{x}{2}) + C \)

Hint:

When dealing with complex trigonometric integrals, substitution and trigonometric identities can simplify the integrand and make the problem easier to solve.

Answer 1

The Correct Option is B

Step 1: Substitution. Let \( u = 1 + \log \left( \tan \frac{x}{2} \right) \). Differentiating both sides: \[ \frac{du}{dx} = \frac{1}{\tan \frac{x}{2}} \cdot \sec^2 \frac{x}{2} \cdot \frac{1}{2}. \] Thus: \[ du = \frac{\csc x}{2} \, dx \quad \Rightarrow \quad 2 du = \frac{\csc x}{\cos^2 u} \, dx. \] Step 2: Substituting into the integral. The integral becomes: \[ I = 2 \int \sec^2 u \, du. \] Step 3: Solving the integral. The integral of \( \sec^2 u \) is \( \tan u \), so we have: \[ I = 2 \tan u + C. \] Step 4: Substitute back the value of \( u \). Substituting \( u = 1 + \log \left( \tan \frac{x}{2} \right) \) into the result: \[ I = 2 \tan \left( 1 + \log \left( \tan \frac{x}{2} \right) \right) + C. \] Since the constant factor of 2 can be simplified, the final result is: \[ I = \tan \left( 1 + \log \left( \tan \frac{x}{2} \right) \right) + C. \]