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$space F i n d space colon integral fraction numerator left parenthesis 2 x minus 5 right parenthesis e to the power of 2 x end exponent over denominator left parenthesis 2 x minus 3 right parenthesis end fraction d x$

Consider the given integral
$straight I space equals integral fraction numerator left parenthesis 2 straight x minus 5 right parenthesis straight e to the power of 2 straight x end exponent over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction dx Rewriting space the space above space integral space as straight I space equals space integral straight e to the power of 2 straight x minus 3 end exponent space straight x space straight e cubed fraction numerator left parenthesis 2 straight x minus 3 minus 2 right parenthesis over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction dx equals space straight e cubed integral straight e to the power of 2 straight x minus 3 end exponent open square brackets fraction numerator left parenthesis 2 straight x minus 3 right parenthesis over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction minus fraction numerator 2 over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction close square brackets dx equals space straight e cubed integral straight e to the power of 2 straight x minus 3 end exponent open square brackets fraction numerator 1 over denominator left parenthesis 2 straight x minus 3 right parenthesis squared end fraction minus fraction numerator 2 over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction close square brackets dx Let space us space consider comma space 2 straight x minus 3 space equals space straight t rightwards double arrow space 2 dx space equals space dt therefore space straight I equals straight e cubed integral straight e to the power of straight t open square brackets 1 over straight t squared minus 2 over straight t cubed close square brackets dt over 2 rightwards double arrow space straight I equals straight e cubed over 2 integral straight e to the power of straight t open square brackets fraction numerator straight t minus 2 over denominator straight t cubed end fraction close square brackets space dt Let space straight f space left parenthesis straight t right parenthesis space equals 1 over straight t squared straight f apostrophe left parenthesis straight t right parenthesis space equals space fraction numerator negative 2 over denominator straight t cubed end fraction If space straight I space equals integral straight e to the power of straight t space end exponent left square bracket straight f space left parenthesis straight t right parenthesis space plus space straight f apostrophe left parenthesis straight t right parenthesis space right square bracket dt space then comma space straight I space equals space space straight e apostrophe straight f space left parenthesis straight t right parenthesis space plus space straight C therefore space straight I space equals straight e cubed over 2 space straight x space straight e to the power of straight t space space straight x space straight f space left parenthesis straight t right parenthesis space plus space straight C equals space straight e cubed over 2 space straight x space space straight e to the power of 2 straight x minus 3 end exponent space space straight x space fraction numerator 1 over denominator left parenthesis 2 straight x minus 3 right parenthesis squared end fraction space plus space straight C equals fraction numerator straight e to the power of 2 straight x end exponent over denominator 2 left parenthesis 2 straight x minus 3 right parenthesis squared end fraction plus straight C$

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Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$

Consider the given differential equation:
$xlogx dy over dx plus straight y space equals space 2 logx$
Dividing the above equation by xlogx, we have,
$space space fraction numerator xlogx space over denominator xlogx end fraction dy over dx space plus space fraction numerator straight y over denominator straight x space logx end fraction space equals space fraction numerator 2 logx over denominator xlogx end fraction rightwards double arrow dy over dx plus Py space equals space straight Q comma space where space straight P space and space straight Q space are space functions space of space straight x Comparing space equation space left parenthesis 1 right parenthesis space and space the space general space equation comma space we space have comma space straight P left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator straight x space logx end fraction space and space straight Q left parenthesis straight x right parenthesis space equals space 2 over straight x The space intergrating space factor space is space given space by space the space formulae space straight e to the power of integral Pdx end exponent space Thus comma space straight I. straight F. space equals space straight e to the power of integral Pdx end exponent space space equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent$
$Consider space straight I space equals space integral dx over xlogx Substituting space logx equals space straight t semicolon space space dx over straight x space equals space dt Thus space straight I space equals space integral dt over straight t space equals space log left parenthesis straight t right parenthesis space equals space log space left parenthesis log space straight x right parenthesis Hence comma space straight I. straight F. equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent space equals space straight e to the power of log left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x$