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Math

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Help with these problems would be greatly appreciated:

1.Find the definite integral of dx/(x(1+ln(x))from e^6 to 1.

2. Solve for x in terms of k for log[2,x^6)+log[2,x^10=k. (its log base 2)

3.Solve log base 3(log base 3, x)=-2

  • Calculus - beyond me -

    Perhaps it is the lateness of the hour but I am stuck on the first one.

  • Math -

    1. For the indefinite integral, try the substitution ln x = u
    e^u = x
    e^u du = dx
    indef. integral of dx/(x(1+ln(x))
    = integral of e^u du/[e^u*(1 + u)
    = integral of du/(1+u)
    = ln (1 + u) = ln (1 + ln x)
    When x = e^6, the indef. integral is ln (1 + 6) = ln 7 = 1.9459
    When x = 1, the indef. integral is ln 1 = 0
    Def. integral = -ln 7


    3. Rewrite as
    3^-2 = log3,x = 1/9
    3^(1/9) = x
    x = 1.129831

  • Math -

    2. log2 x^6 + log2 x^10 = k
    6log2x + 10log2x = k
    16log2x = k
    log2x^16 = k
    2^k = x^16
    x = 2^(k/16)

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