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Math Help - more integration help please

  1. #1
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    more integration help please

    how about this one: integrand (2 + 2sinx)^(1/2) dx? The book says "We could evaluate this integral by multiplying and dividing the integrand by (2-2sinx)^(1/2) Do you use partial integration? It's from Stewart's Early Transcendentals fifth edition page 683
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  2. #2
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    Re: more integration help please

    I don't know what you mean by "partial integration". integration by "partial fractions"? Have you tried doing what it suggests? Once you multiply both numerator and denominator by (2- 2sin(x))^{1/2} you will have \int\frac{2cos(x)dx}{(2- 2sin(x))^{1/2}}=  \sqrt{2}\int\frac{cos(x)dx}{(1- sin(x))^{1/2}}} and the substitution u= 1- sin(x) looks like something to try.
    Thanks from topsquark
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  3. #3
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    Re: more integration help please

    Quote Originally Posted by HallsofIvy View Post
    I don't know what you mean by "partial integration". integration by "partial fractions"? Have you tried doing what it suggests? Once you multiply both numerator and denominator by (2- 2sin(x))^{1/2} you will have \int\frac{2cos(x)dx}{(2- 2sin(x))^{1/2}}=  \sqrt{2}\int\frac{cos(x)dx}{(1- sin(x))^{1/2}}} and the substitution u= 1- sin(x) looks like something to try.
    I honestly did try. But I didn't get what you got. But now that I see it I realize my mistake. Sometimes I forget that as long as they're both under the radical sign you can combine terms underneath one radical sign. Thanks for the help!
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  4. #4
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    Re: more integration help please

    This is actually an arc length problem for the polar equation r=1+sin(x) (x being theta) After some work you get:
    L=integral from 0 to 2 times pi (2+2*sin(x))^(1/2) dx......as above it gets reduced to

    L=2^(1/2) times integral from 0 to 2 times pi (cos(x)/(1-sin(x))^1/2) dx

    But if u=1-sin(x) don't the limits of integration both equal 1?

    The book says the answer is L=8 The figure is a cardioid.
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