Question

If $ I_n = \int_0^\pi \tan^n x \, dx $, for $ n \geq 2 $, then $ I_n + I_{n-2} = $

• A. \( \frac{n}{n-1} \)
• B. \( \frac{1}{n+1} \)
• C. \( \frac{1}{n} + \frac{1}{n-2} \)
• D. \( \frac{1}{n} \)

Hint:

In solving such problems, use known identities for integrals involving powers of trigonometric functions like \( \tan x \).

Answer 1

The Correct Option is A

Using known results from the properties of integrals involving trigonometric functions, for even values of \( n \), we have: \[ I_n + I_{n-2} = \frac{n}{n-1} \]