Question

If \( I_a = \int_0^{\pi/4} \tan^n x \, dx \), then \[ \frac{1}{I_2 + I_4} + \frac{1}{I_3 + I_5} + \frac{1}{I_4 + I_6} = \]

• A. \( \frac{1}{I_9 + I_{11}} \)
• B. \( \frac{1}{I_{10} + I_{12}} \)
• C. \( \frac{1}{I_{12} + I_{14}} \)
• D. \( \frac{1}{I_{11} + I_{13}} \)

Hint:

Memorize or derive integral identities for powers of tangent to relate higher order integrals.

Answer 1

The Correct Option is D

This is based on recurrence relations of integrals of the form \( I_n = \int \tan^n x \, dx \). Use reduction formulas or patterns for odd and even powers, then combine the identities to simplify.