Question

Evaluate the integral \[ \int \frac{\sec x}{\sqrt{\log(\sec x + \tan x)}}\, dx \]

• A. \( \sqrt{\log(\sec x + \tan x)} + c \)
• B. \( \sqrt{\sec x + \tan x} + c \)
• C. \( 2\sqrt{\sec x + \tan x} + c \)
• D. \( 2\sqrt{\log(\sec x + \tan x)} + c \)

Hint:

Whenever an integrand contains \( \sec x \) along with \( \sec x + \tan x \), try the substitution \( u = \log(\sec x + \tan x) \).

Answer 1

The Correct Option is D

Step 1: Choose an appropriate substitution.
Let \[ u = \log(\sec x + \tan x) \]

Step 2: Differentiate with respect to \( x \).
\[ \frac{du}{dx} = \frac{\sec x (\tan x + \sec x)}{\sec x + \tan x} = \sec x \] \[ du = \sec x \, dx \]

Step 3: Substitute into the integral.
\[ \int \frac{\sec x}{\sqrt{\log(\sec x + \tan x)}}\, dx = \int \frac{du}{\sqrt{u}} \]

Step 4: Integrate and back-substitute.
\[ \int u^{-1/2} du = 2u^{1/2} + c \] \[ = 2\sqrt{\log(\sec x + \tan x)} + c \]

Step 5: Conclusion.
\[ \boxed{2\sqrt{\log(\sec x + \tan x)} + c} \]