Question

If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)

• A. 6
• B. 5
• C. 4
• D. 3

Hint:

Compare series indices and match terms on both sides carefully to identify bounds.

Answer 1

The Correct Option is B

We are told:
\[ I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \] Now compare term-wise with the RHS. The sum on the right has \( n \) terms from \( k = 1 \) to \( k = n \).
The presence of \( \frac{\tan^5 x}{5} \) means \( n = 5 \) is the last term on the right.
Hence, the correct value of \( n \) is \( \boxed{5} \).