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INTEGRATE  Sin(3x/2)÷sin (x/2) limit varying from 90° to 180° with respect to X

Asked by hemanttkumarr 12th January 2019, 7:35 AM
Answered by Expert
Answer:
begin mathsize 16px style I space equals space integral subscript bevelled straight pi over 2 end subscript superscript straight pi fraction numerator sin open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator sin open parentheses begin display style x over 2 end style close parentheses end fraction d x
l e t space x over 2 equals space theta space semicolon space d x space equals space 2 d theta
I space equals space 2 integral subscript straight pi superscript 2 straight pi end superscript fraction numerator sin begin display style space end style begin display style 3 end style begin display style theta end style over denominator sin begin display style space end style begin display style theta end style end fraction d theta
sin space 3 theta space equals space sin left parenthesis 2 theta plus theta right parenthesis space equals space sin 2 theta cos theta space plus space cos 2 theta sin theta space equals space 2 sin theta cos squared theta plus cos 2 theta sin theta
fraction numerator sin begin display style 3 end style begin display style theta end style over denominator sin begin display style theta end style end fraction space equals space 2 cos squared theta plus cos 2 theta space equals space 1 plus 2 cos 2 theta
I space equals 2 integral subscript straight pi superscript 2 straight pi end superscript d theta space plus space 2 integral subscript 2 straight pi end subscript superscript 4 straight pi end superscript cos 2 theta space d left parenthesis 2 theta right parenthesis
s e c o n d space i n t e g r a l space v a n i s h e s comma space b e c a u s e space cos i n e space f u n c t i o n space i n t e g r a t e d space
o n e space f u l l p e r i o d space left parenthesis i e. space 2 pi space t o space 4 pi right parenthesis
I space equals space 2 open parentheses 2 straight pi minus straight pi close parentheses space equals space 2 straight pi
end style
Answered by Expert 12th January 2019, 10:41 AM
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