Question

Evaluate the integral \[ \int \frac{(1 + \log x)}{\cos^2(\log x)} \, dx \]

• A. \( \sin(\log x) + c \)
• B. \( \sin^2(\log x) + c \)
• C. \( \log(\log x) + c \)
• D. \( \tan(\log x) + c \)

Hint:

For integrals involving logarithmic and trigonometric functions, try substitution to simplify the expression.

Answer 1

The Correct Option is D

Step 1: Use substitution.
Let \( u = \log x \). Then, \( du = \frac{1}{x} dx \), and the integral becomes: \[ \int \frac{(1 + u)}{\cos^2 u} \, du \]
Step 2: Solve the integral.
The integral simplifies to: \[ \int (1 + u) \sec^2 u \, du = \tan u + c = \tan(\log x) + c \]
Step 3: Conclusion.
Thus, the integral evaluates to \( \tan(\log x) + c \), corresponding to option (D).