Question:
What is \(\int_a^b [x]\,dx + \int_a^b [-x]\,dx\) equal to, where \([.]\) is the greatest integer function?
\(\int_a^b [x]\,dx + \int_a^b [-x]\,dx\) किसके बराबर है, जहाँ \([.]\) महत्तम पूर्णांक फलन है?
Options:
- \(b – a\)
- \(a – b\)
- 0
- \(2(b – a)\)
- \(b – a\)
- \(a – b\)
- 0
- \(2(b – a)\)
Correct Answer: b)
Explanation:
We know that for any real number \(x\):
\[[x] + [-x] = \begin{cases} 0, & \text{if } x \in \mathbb{Z} \\ -1, & \text{if } x \notin \mathbb{Z} \end{cases}\]
Since the set of integers \(\mathbb{Z}\) has measure zero, the integrand is \(-1\) almost everywhere on \([a, b]\):
\[\int_a^b ([x] + [-x])\,dx = \int_a^b (-1)\,dx = -[x]_a^b = -(b – a) = a – b\]
Hence, the correct option is (b).
हम जानते हैं कि सभी गैर-पूर्णांक \(x\) के लिए \([x] + [-x] = -1\) होता है।
पूर्णांकों का समुच्चय शून्य माप (measure zero) का होता है, अतः समाकल:
\[\int_a^b ([x] + [-x])\,dx = \int_a^b (-1)\,dx = -(b – a) = a – b\]
अतः सही विकल्प (b) है।