Question
Download Solution PDFनिम्नलिखित का मूल्यांकन करें:
\(\smallint \frac{{{\rm{cos}}\left( {{\rm{ln}}\left( {\rm{x}} \right)} \right)}}{{\rm{x}}}{\rm{dx}}\)Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFधारणा:
1. प्रतिस्थापन द्वारा समाकलन:
- यदि दिया गया समाकलन \(\smallint {\rm{g}}\left( {{\rm{f}}\left( {\rm{x}} \right)} \right){\rm{f'}}\left( {\rm{x}} \right){\rm{dx}}\) रूप का है जहाँ g(x) और f(x) दोनों अवकलनीय फलन हैं तो हम f(x) = u को प्रतिस्थापित करते हैं जिसका अर्थ है f’ (x)dx = du।
- इसलिए, समाकल \(\smallint {\rm{g}}\left( {\rm{u}} \right){\rm{du}}\) बन जाता है जिसे सामान्य सूत्रों द्वारा हल किया जा सकता है।
समाधान:
दी गई समस्या में \(\ln \left( {\rm{x}} \right) = u\) प्रतिस्थापित करें इसलिए \(\frac{{{\rm{dx}}}}{{\rm{x}}} = {\rm{du}}\)।
दिया गया समाकल बन जाता है,
\(\smallint \cos {\rm{udu}} = \sin {\rm{u}} + {\rm{C}}\)
\({\rm{u}} = \ln \left( {\rm{x}} \right)\) को पुनःप्रतिस्थापित करें।
\(\smallint \frac{{\cos \left( {\ln {\rm{x}}} \right)}}{{\rm{x}}}{\rm{dx}} = \sin \left( {\ln {\rm{x}}} \right) + {\rm{C}}\)Last updated on Jun 30, 2025
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