calculus

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Evaluate the following
(x+7)dx/x^2+14x+55

  • calculus - eh? -

    makes no sense as written
    better look again and copy it accurately

  • calculus -

    If (x+7)dx/x^2+14x+55

    mean

    ( x + 7 ) dx / ( x ^ 2 + 14 x + 55 )

    then substitute :

    u = x ^ 2 + 14 x + 5

    d u = ( 2 x + 14 ) dx = 2 ( x + 7 ) d x Divide both sides by 2

    du / 2 = ( x + 7 ) d x

    ( x + 7 ) d x = d u / 2


    integral of [ ( x + 7 ) dx / ( x ^ 2 + 14 x + 55 ) ] =

    integral of [ ( d u / 2 ) / u ) ] =

    ( 1 / 2 ) integral of ( du / u ) =

    ( 1 / 2 ) log ( u ) + C =

    ( 1 / 2 ) log ( x ^ 2 + 14 x + 5 ) + C =

    log [ sqrt ( x ^ 2 + 14 x + 5 ) ] + C


    Remark :

    log is the natural logarithm

    ( 1 / n ) log ( a ) = log ] ( n - th root ( a ) ]

    ( 1 / 2 ) log ( a ) = log [ sqrt ( a ) ]

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