Question

Solve:
$dy over dx equals negative straight x space sin squared straight x space space equals space fraction numerator 1 over denominator straight x space log space straight x end fraction$

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Solution

dy over dx minus straight x space sin squared straight x space equals space fraction numerator 1 over denominator straight x space log space straight x end fraction space or space dy over dx space equals straight x space sin squared straight x space plus space fraction numerator 1 over denominator straight x space logx end fraction

Separating the variables, we get,

dy equals open parentheses straight x space sin squared straight x plus fraction numerator 1 over denominator straight x space logx space end fraction close parentheses dx space space space space rightwards double arrow space space integral dy space equals space integral open parentheses xsin squared straight x plus fraction numerator 1 over denominator straight x space logx end fraction close parentheses dx

therefore space space space integral space dy space equals integral straight x space sin squared straight x space dx plus integral fraction numerator 1 over denominator straight x space logx space dx end fraction therefore space space space integral dy space equals space straight I subscript 1 plus straight I subscript 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

where

straight I subscript 1 space equals space integral straight x space sin squared space dx space equals space 1 half integral straight x space left parenthesis 2 space sin squared straight x right parenthesis space dx

equals space 1 half integral straight x left parenthesis 1 minus cos space 2 straight x right parenthesis space dx space equals space 1 half integral straight x space dx space minus space 1 half integral straight x. space cos space 2 straight x space dx

equals space straight x squared over 4 minus 1 fourth straight x space sin space 2 straight x space plus space 1 fourth integral sin space 2 straight x space dx space equals space straight x squared over 4 minus 1 fourth straight x space sin 2 straight x space minus 1 over 8 cos space 2 straight x

straight I subscript 2 space equals integral fraction numerator 1 over denominator straight x space logx end fraction dx space equals space integral fraction numerator begin display style 1 over straight x end style over denominator log space straight x end fraction dx space equals space log space open vertical bar log space straight x close vertical bar

therefore space space space from space left parenthesis 1 right parenthesis comma space we space get

integral dy space equals space straight x squared over 4 minus 1 fourth straight x space sin space 2 straight x space minus 1 over 8 cos space 2 straight x space plus space log space open vertical bar log space straight x close vertical bar

therefore space space space straight y space equals space straight x squared over 4 minus 1 fourth straight x space sin space 2 straight x space minus space 1 over 8 cos space 2 straight x space plus space log space open vertical bar logx close vertical bar space plus straight c

which is the required solution.

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