Question
Download Solution PDFयदि \(f\left( x \right) = \left[ {\frac{1}{x}} \right]\) है, जहाँ [.] सबसे बड़ा पूर्णांक फलन है। तो \(\mathop \smallint \nolimits_{\frac{1}{3}}^{\frac{1}{2}} f\left( x \right)\;dx\) का मान ज्ञात कीजिए।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
सबसे बड़ा पूर्णांक फलन: (फ्लोर फलन)
फलन f (x) = [x] को सबसे बड़ा पूर्णांक फलन कहा जाता है और इसका अर्थ है कि सबसे बड़ा पूर्णांक x से कम या इसके बराबर है अर्थात् [x] ≤ x है।
[x] का डोमेन R है और सीमा I है।
गणना:
दिया गया है: \(f\left( x \right) = \left[ {\frac{1}{x}} \right]\)
दिए गए समाकल \(\mathop \smallint \nolimits_{\frac{1}{3}}^{\frac{1}{2}} f\left( x \right)\;dx\) के लिए, जब x, 1 / 3 और 1 / 2 अर्थात्\(\frac{1}{3} < x < \frac{1}{2}\) के बीच में होता है।
\(\Rightarrow 2 < \frac{1}{x} < 3\)
इसलिए, उपरोक्त असमानता के अनुसार: \(f\left( x \right) = \left[ {\frac{1}{x}} \right] = 2\)
\(\Rightarrow \mathop \smallint \nolimits_{\frac{1}{3}}^{\frac{1}{2}} f\left( x \right)\;dx = \;\mathop \smallint \nolimits_{\frac{1}{3}}^{\frac{1}{2}} 2\;dx = 2 \times \left( {\frac{1}{2} - \frac{1}{3}} \right) = \frac{1}{3}\)Last updated on Jul 7, 2025
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