If I_ n = _ 0 ^ 1 d x (1+x^ 2 )^ n , n N, then which of the following statements hold…
Mathematics · JEE Advanced · NTA Exams — Integral Calculus
If \(I_{n}=\int_{0}^{1} \frac{d x}{\left(1+x^{2}\right)^{n}}, n \in N,\) then which of the following statements hold good ?
- \(2 n I_{n+1}=2^{-n}+(2 n-1) I_{n}\)
- \(I_{2}=\frac{\pi}{8}+\frac{1}{4}\)
- \(I_{2}=\frac{\pi}{8}-\frac{1}{4}\)
- \(I_{3}=\frac{\pi}{16}-\frac{5}{48}\)
Answer
(A) 2 n I_ n+1 =2^ -n +(2 n-1) I_ n, (B) I_ 2 = 8 + 1 4
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