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Sir pls solve this question.

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Asked by rsudipto 11th December 2018, 9:34 AM
Answered by Expert
Answer:
begin mathsize 12px style I f space fraction numerator d over denominator d x end fraction open curly brackets g open parentheses x close parentheses close curly brackets space equals space g left parenthesis x right parenthesis space space space comma space t h e n space integral g left parenthesis x right parenthesis d x space equals space g left parenthesis x right parenthesis end style, also we are given g(0) = 1
Exponetial function ex satisfies the above condition, But without using exponential function let us solve this question
 
we have begin mathsize 12px style fraction numerator 2 space minus space sin 2 x over denominator 1 space minus space cos 2 x end fraction space equals space fraction numerator 2 minus 2 space sin x space cos x over denominator 2 space sin squared x end fraction space equals space cos e c squared left parenthesis x right parenthesis space minus space c o t space left parenthesis x right parenthesis end style
hence begin mathsize 12px style I space equals integral g left parenthesis x right parenthesis fraction numerator 2 minus sin 2 x over denominator 1 minus cos 2 x end fraction space d x space equals space integral g left parenthesis x right parenthesis open parentheses cos e c squared x space minus space c o t x close parentheses d x space equals space I subscript 1 space minus space I subscript 2 space plus space C
w h e r e space I subscript 1 equals space integral g left parenthesis x right parenthesis space cos e c squared x space d x space space space a n d space space I subscript 2 space equals space integral g left parenthesis x right parenthesis space c o t left parenthesis x right parenthesis space d x
L e t space u s space e x a m i n e space I subscript 2 space colon space space
I subscript 2 space equals space integral g left parenthesis x right parenthesis space c o t left parenthesis x right parenthesis space d x space
p u t space u space equals space c o t left parenthesis x right parenthesis space a n d space d v equals space g left parenthesis x right parenthesis d x space semicolon space t h e n space v space equals space integral g left parenthesis x right parenthesis d x space equals space g left parenthesis x right parenthesis space a n d space d u space equals space minus cos e c squared x space d x

H e n c e space I subscript 2 space equals space g left parenthesis x right parenthesis space c o t space left parenthesis x right parenthesis space plus integral g left parenthesis x right parenthesis space cos e c squared x space d x space equals space g left parenthesis x right parenthesis space c o t space left parenthesis x right parenthesis space plus space I subscript 1

H e n c e space I space equals space I subscript 1 minus I subscript 2 space plus C equals space I subscript 1 space minus space g left parenthesis x right parenthesis space c o t space left parenthesis x right parenthesis space minus space I subscript 1 space plus space C equals space minus g left parenthesis x right parenthesis space c o t space left parenthesis x right parenthesis space plus C end style
Answered by Expert 13th December 2018, 5:04 AM
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