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JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f:(0,2) \rightarrow R\) be defined as \(f( x )=\log _{2}\left(1+\tan \left(\frac{\pi x }{4}\right)\right)\) Then, \(\lim _{n \rightarrow \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots+f(1)\right)\) is equal to
- A \(2\)
- B \(1\)
- C \(4\)
- D \(5\)
Answer & Solution
Correct Answer
(B) \(1\)
Step-by-step Solution
Detailed explanation
\(E =2 \lim _{ n \rightarrow \infty} \sum_{ r =1}^{ n } \frac{1}{ n } f \left(\frac{ r }{ n }\right)\) \(E =\frac{2}{\ell n 2} \int_{0}^{1} \ell n \left(1+\tan \frac{\pi x }{4}\right) dx\) \(.........(i)\) replacing \(x \rightarrow 1- x\)…
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