Verified Solution • 2025–26 Session moderate

QJEEM21Ma8697: JEE Main 2021 Mathematics - Class 12 JEEM21Ma8697 Solution: The value of , where [ x ] is the greatest integer x, is :

The value of ∑n=1100∫n−1nex−[x]dx\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} } , where [ x ] is the greatest integer ≤\le x, is :

Multiple Choice Options

  • A 100e
  • B 100(e −- 1)
  • C 100(1 + e)
  • D 100(1 −- e)

Step-by-Step Verified Solution

∑n=1100∫n−1nex−[x]dx\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}} dx}

Here, n - 1 \le x < n

∴\therefore [x]=n−1[x] = n - 1

∴\therefore ∫n−1nex−(n−1)dx\int\limits_{n - 1}^n {{e^{x - (n - 1)}}} dx

=[ex−(n−1)]n−1n= \left[ {{e^{x - (n - 1)}}} \right]_{n - 1}^n

=e1−e0= {e^1} - {e^0}

=e−1= e - 1

Now, ∑n=1100(e−1)=100(e−1)\sum\limits_{n = 1}^{100} {(e - 1) = 100(e - 1)}

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