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CBSE Class 12-science Answered

Sir Question no-4
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Asked by shobhit | 22 Feb, 2019, 11:31: PM
answered-by-expert Expert Answer
Numerator integration
begin mathsize 12px style I subscript n r space end subscript equals integral subscript 0 superscript bevelled straight pi over 2 end superscript open parentheses x space cos x space plus 1 close parentheses space e to the power of sin space x end exponent space d x space equals space integral subscript 0 superscript bevelled straight pi over 2 end superscript x space cos x space e to the power of sin space x end exponent space d x space plus space integral subscript 0 superscript bevelled straight pi over 2 end superscript e to the power of sin space x end exponent space d x space equals space I subscript n r 1 end subscript space plus space I subscript n r 2 end subscript I subscript n r 2 end subscript space equals space space integral subscript 0 superscript bevelled straight pi over 2 end superscript e to the power of sin space x end exponent space d x L e t space u space equals space e to the power of sin space x end exponent space space space comma space space d u space equals space e to the power of sin x end exponent space cos x space d x space space space semicolon space space space d v space equals space d x space comma space space space v space equals space x I subscript n r 2 end subscript space equals open parentheses x space e to the power of sin space x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space space minus integral subscript 0 superscript bevelled straight pi over 2 end superscript x space cos x space e to the power of sin space x end exponent space d x space space equals space open parentheses x space e to the power of sin space x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space space minus I subscript n r 1 end subscript I subscript n r space end subscript equals space I subscript n r 1 end subscript space plus space I subscript n r 2 end subscript space equals space open parentheses x space e to the power of sin space x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space equals straight pi over 2 e end style
 
 
Denominator Integration
begin mathsize 12px style I subscript d r space end subscript equals integral subscript 0 superscript bevelled straight pi over 2 end superscript open parentheses x space sin x space minus 1 close parentheses space e to the power of cos space x end exponent space d x space equals space integral subscript 0 superscript bevelled straight pi over 2 end superscript x space sin x space e to the power of cos space x end exponent space d x space minus space integral subscript 0 superscript bevelled straight pi over 2 end superscript e to the power of cos space x end exponent space d x space equals space I subscript d r 1 end subscript space minus space I subscript d r 2 end subscript I subscript d r 2 end subscript space equals space space integral subscript 0 superscript bevelled straight pi over 2 end superscript e to the power of cos space x end exponent space d x L e t space u space equals space e to the power of cos space x end exponent space space space comma space space d u space equals space minus e to the power of cos x end exponent space sin x space d x space space space semicolon space space space d v space equals space d x space comma space space space v space equals space x I subscript d r 2 end subscript space equals open parentheses x space e to the power of cos space x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space space plus integral subscript 0 superscript bevelled straight pi over 2 end superscript x space sin x space e to the power of cos space x end exponent space d x space space equals space open parentheses x space e to the power of cos x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space space plus I subscript d r 1 end subscript I subscript d r space end subscript equals space I subscript d r 1 end subscript space minus space I subscript d r 2 end subscript space equals space minus open parentheses x space e to the power of cos x end exponent space space close parentheses subscript 0 superscript bevelled straight pi over 2 end superscript space equals negative straight pi over 2 e end style
 
Hence begin mathsize 12px style open vertical bar I subscript n r end subscript over I subscript d r end subscript close vertical bar space equals space open vertical bar fraction numerator begin display style straight pi over 2 end style e over denominator negative begin display style straight pi over 2 end style end fraction close vertical bar space space equals space e end style
Answered by Thiyagarajan K | 23 Feb, 2019, 08:32: AM

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