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Integrals

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

1225 Views

Find space colon integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx

Taking space the space given space function straight I equals integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx Let space fraction numerator straight x squared space plus space straight x space plus space 1 over denominator left parenthesis straight x squared space plus space 1 right parenthesis space left parenthesis straight x space plus space 2 right parenthesis end fraction space space equals space fraction numerator straight A over denominator straight x space plus space 2 end fraction space plus space fraction numerator Bx space plus space straight C over denominator straight x squared plus 1 end fraction equals space fraction numerator straight A left parenthesis straight x squared space plus space 1 right parenthesis space plus space left parenthesis Bx space space plus space straight C right parenthesis left parenthesis straight x space plus space 2 right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis straight A plus straight B right parenthesis straight x squared plus left parenthesis 2 straight B plus straight C right parenthesis space straight x plus left parenthesis 2 straight C plus straight A right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction Thus space equating space the space coefficients comma space we space have comma straight A space plus space straight B space equals 1 space.. space left parenthesis straight i right parenthesis 2 straight B space space plus space straight C space equals space 1 space.. left parenthesis ii right parenthesis 2 straight C space plus space straight A space equals space 1 space.. space left parenthesis iii right parenthesis solving space the space above space three space equations comma space we space have comma straight A space equals 3 over 5 comma space straight B space equals 2 over 5 space and space straight C equals 1 fifth therefore comma fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction space fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator straight x squared plus 1 end fraction rightwards double arrow space fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction space equals space fraction numerator 3 over denominator 5 left parenthesis straight x plus 2 right parenthesis end fraction space plus space fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction therefore comma straight I equals integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx equals integral open square brackets fraction numerator 3 over denominator 5 space left parenthesis straight x plus 2 right parenthesis end fraction plus fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction dx close square brackets equals 3 over 5 integral fraction numerator dx over denominator left parenthesis straight x plus 2 right parenthesis end fraction dx space plus space 1 fifth integral fraction numerator 2 straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx space plus 1 fifth integral fraction numerator dx over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth log space left parenthesis straight x squared plus 1 right parenthesis plus 1 fifth tan to the power of negative 1 end exponent space straight x space plus space straight C

1137 Views

Find colon space integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared space end root space dx.

straight I space equals integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx Let space left parenthesis straight x space plus 3 right parenthesis space equals space straight A straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis space plus space straight B space space space... space left parenthesis straight i right parenthesis rightwards double arrow space space straight x space plus space 3 space equals space straight A left parenthesis negative 2 straight x minus 4 right parenthesis space plus space straight B rightwards double arrow space straight x plus 3 space equals space minus 2 Ax minus 4 straight A plus straight B therefore space minus 2 straight A space space equals space 1 rightwards double arrow space straight A space equals space minus 1 half minus 4 straight A space plus space straight B space equals space 3 rightwards double arrow space minus 4 space open parentheses negative 1 half close parentheses plus space straight B space equals 3 rightwards double arrow space space straight B equals 1 therefore comma straight I space equals integral open square brackets negative 1 half straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis plus 1 close square brackets space square root of 3 minus 4 straight x minus straight x squared end root space dx equals space space 1 half integral straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx space plus space integral square root of 3 minus 4 straight x minus straight x squared minus 4 plus 4 dx end root equals space minus 1 half fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 divided by 2 end fraction space plus space integral square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root dx equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared dx end root equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root space plus space 7 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator square root of 7 end fraction close parentheses space space plus space straight C

693 Views

Evaluate space colon space integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx.

Consider space the space given space integral straight I equals integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space space.... left parenthesis straight i right parenthesis Let space us space use space the space property comma integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals integral subscript straight a superscript straight b straight f left parenthesis straight a plus straight b minus straight x right parenthesis dx therefore space straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator left parenthesis negative straight x right parenthesis squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx equals integral subscript negative 2 end subscript superscript 2 fraction numerator 5 to the power of straight x space straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space.. space left parenthesis ii right parenthesis Adding space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have comma 2 straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator 1 plus 5 to the power of straight x over denominator 1 plus 5 to the power of straight x end fraction space. space straight x squared dx equals integral subscript negative 2 end subscript superscript 2 straight x squared dx equals space open square brackets straight x cubed over 3 close square brackets subscript negative 2 end subscript superscript 2 equals space 1 third left square bracket 8 minus left parenthesis negative 8 right parenthesis right square bracket equals space 1 third space left square bracket 16 right square bracket rightwards double arrow space straight I space equals space 8 over 3

799 Views

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

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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Integrals

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