Question

Evaluate the integral: \[ \int \frac{dx}{4 + 5 \cos x} \]

• A. \( -\frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)
• B. \( \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)
• C. \( -\frac{1}{9} \log \left| \frac{3 - \tan \frac{x}{2}}{3 + \tan \frac{x}{2}} \right| + C \)
• D. \( -\frac{1}{9} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)

Hint:

For integrals involving trigonometric functions, try substituting \( t = \tan \frac{x}{2} \) to simplify the expression and use standard results for the integral.

Answer 1

The Correct Option is B

The given integral is \( \int \frac{dx}{4 + 5 \cos x} \). Step 1: Use the substitution \( t = \tan \frac{x}{2} \), which simplifies the trigonometric expression. Step 2: The integral becomes: \[ \int \frac{dx}{4 + 5 \cos x} = \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \] % Final Answer The correct integral is \( \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \).