Maths · Evaluation of definite integrals
If then are in?
If \( \boldsymbol{I}_{n}=\int_{0}^{\frac{\pi}{4}} \tan ^{n} x d x \) then \( \frac{1}{I_{2}+I_{4}}, \frac{1}{I_{3}+I_{5}}, \frac{1}{I_{4}+I_{6}} \) are in?
- A. \( A . P \)
- B. \( H . P \)
- C. \( G . P \)
- D. None of these
Step-by-step solution
Using the recurrence relation I_n + I_{n-2} = 1/(n-1) derived from integrating tan^n x = tan^{n-2}x sec^2 x - tan^{n-2}x, we get I_2 + I_4 = 1/3, I_3 + I_5 = 1/4, I_4 + I_6 = 1/5. Thus, the given terms are 3, 4, 5, which are in arithmetic progression.