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Math Help - Converge or Diverge?

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    Converge or Diverge?

    Does the integral  \int_{1}^{\infty} \frac{3 \; dx}{2x \sqrt{4+5x}} converge or diverge?

    If it converges, what is its value?
    Last edited by shilz222; September 25th 2007 at 10:22 PM.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    Does the integral  \int_{1}^{\infty} \frac{3x \; dx}{2x \sqrt{4+5x}} converge or diverge?

    If it converges, what is its value?
    \int_{1}^{\infty} \frac{3x}{2x \sqrt{4+5x}}~dx = \frac 32 \lim_{N \to \infty} \int_{1}^{N} \frac 1{\sqrt {4 + 5x}}~dx

    We proceed by substitution:

    Let u = 4 + 5x

    \Rightarrow du = 5 ~dx

    \Rightarrow \frac 15~du = dx

    So our integral becomes:

    \frac 3{10} \lim_{N \to \infty} \int_{x = 1}^{x = N} u^{- \frac 12}~du = \frac 3{10} \left[ 2 u^{\frac 12} \right]_{x = 1}^{x = N}

    = \frac 3{10} \lim_{N \to \infty} \left[ 2 \sqrt {4 + 5x} \right]_{1}^{N}

    I leave the rest to you
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    Its  \frac{3}{2} \int \frac{dx}{x \sqrt{4+5x}} . You left out the  x in the denominator.
    what are you talking about? you have 3x in the top and 2x in the bottom. the x's cancel
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    It should be just a 3 in the numerator.
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