Let [x] denote the largest integer not exceeding x and {x} = x - [x]. Then \int^{2012}_{0}{e^{\cos(\pi\{x\})}}/{e^{\cos(\pi\{x\})}+e^{-\cos(\pi\{x\})}}dx is equal to

KVPY SB SX Exam 2011 Question Paper

Question : 15 of 120

Marks: +1, -0

Let [x] denote the largest integer not exceeding x and {x} = x - [x]. Then

∫↖{2012}↙{0}{e^{\cos(π\{x\})}}/{e^{\cos(π\{x\})}+e^{-\cos(π\{x\})}}dx

is equal to

Solution:

I=2012∫↖{1}↙{0}{e^{\cos(π x )}}/{e^{\cos(π x )}+e^{-\cos(π x )}}dx

using king property

I=2012∫↖{1}↙{0}{e^{-\cos(π x )}}/{e^{-\cos(π x )}+e^{ \cos(π x )}}dx

⇒2I=2012⇒I=1006

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