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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
Let $[x]$ denote the largest integer not exceeding $x$ and $\{x\}=x-[x]$. Then, $\int \limits_0^{2012} \frac{e^{\cos (\pi\{x\})}}{e^{\cos (\pi\{x\})}+e^{-\cos (\pi\{x\})}} d x$ is equal to
  • A
    $0$
  • ✓
    $1006$
  • C
    $2012$
  • D
    $2012\,\pi$
✓

Answer

Correct option: B.
$1006$
b
(b)

Explanation :-

$I =2012 \int \limits_0^1 \frac{e^{\cos \pi x}}{e^{\cos \pi x}+e^{-\cos \pi x}} d x$

using king property

$I =2012 \int \limits_0^1 \frac{e^{-\cos \pi x}}{e^{-\cos \pi x}+e^{\cos \pi x}} d x$

$\Rightarrow 2 I =2012 \Rightarrow I =1006$

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