AP EAMCET · Maths · Definite Integration
If \(I_n=\int_0^{\pi / 4} \tan ^n x d x\), then \(I_2+I_4, I_3+I_5, I_4+I_6, \ldots\), are in
- A arithmetic progression
- B geometric progression
- C harmonic progression
- D arithmetico-geometric progression
Answer & Solution
Correct Answer
(C) harmonic progression
Step-by-step Solution
Detailed explanation
\(I_n=\int_0^{\pi / 4} \tan ^n x d x\) We have, \(I_{r+2}=\int_0^{\pi / 4} \tan ^{r+2} x d x\) \(=\int_0^{\pi / 4} \tan ^r x \cdot \tan ^2 x d x\) and \(\quad I_r=\int_0^{\pi / 4} \tan ^r x d x\) Then, \(\quad I_r+I_{r+2}=\int_0^{\pi / 4} \tan ^r x d x\)…
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