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Math Help - integration using ln

  1. #1
    melissah
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    integration using ln

    Studying for my math final, and I'm having some problems with some integration questions... the first:
    Integrate (bottom limit of 2, upper limit of 4) (3/x)dx
    this doesn't look hard, and apparently the answer is 3ln2 but I have no idea how to get there.

    The second is:
    Integrate (bottom limit 1, upper limit 2) [dt/(8-3t)]
    And the answer is (1/3)ln(5/2)

    anyone wanna give me a hand with this?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by melissah View Post
    Studying for my math final, and I'm having some problems with some integration questions... the first:
    Integrate (bottom limit of 2, upper limit of 4) (3/x)dx
    this doesn't look hard, and apparently the answer is 3ln2 but I have no idea how to get there.

    The second is:
    Integrate (bottom limit 1, upper limit 2) [dt/(8-3t)]
    And the answer is (1/3)ln(5/2)

    anyone wanna give me a hand with this?
    \int_2^4 dx \frac{3}{x} = 3 \int_2^4 \frac{dx}{x}

    = 3 (ln(4) - ln(2)) = 3 ln \left ( 2^2 \right ) - 3 ln(2)

    = 3 \cdot 2 ln(2) - 3 ln(2) = 3 ln(2)

    ================================================== ==

    \int_1^2 \frac{dt}{8 - 3t}

    Let x = 8 - 3t. Then dx = -3 dt so
    \int_1^2 \frac{dt}{8 - 3t} = \int_5^2 dx \frac{-1}{3} \frac{1}{x}

    = -\frac{1}{3}(ln(2) - ln(5)) = \frac{1}{3}(ln(5) - ln(2))

    = \frac{ln(5/2)}{3}

    -Dan
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