Question

If \( u_n = \int_0^{\frac{\pi}{4}} \tan \theta d\theta \), then \( u_{n+2} \) is:

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

Hint:

Standard integrals like \( \int \tan \theta \, d\theta \) can be used to simplify the expression for \( u_n \).

Answer 1

The Correct Option is B

The integral for \( u_n \) is a standard trigonometric integral and can be evaluated by standard integration techniques.
Final Answer: \[ \boxed{\frac{1}{n+1}} \]