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Thread: Trig substitutions for integrals

  1. #1
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    Trig substitutions for integrals

    Hi,

    Does anybody know good substitutions to use for integrating:

    x^2/(4-x^2)^1/2

    and

    ((4-x^2)^1/2)/x^2

    I've been thinking I should use u = sin^-1(x/2) for the first and u = cos^-1(x/2) for the second, but these do not seem to yield happy integrals. The first yields integral of 4sin^2(u) du (which is doable but not especially neat using a cos(2u) substitution), and the second yields integral of -tan^2(u) du, which I do not know how to do (but maybe I should?). Are there better substitutions than these?

    Cheers
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  2. #2
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    Quote Originally Posted by DangerousDave View Post
    Hi,

    Does anybody know good substitutions to use for integrating:

    x^2/(4-x^2)^1/2

    and

    ((4-x^2)^1/2)/x^2

    I've been thinking I should use u = sin^-1(x/2) for the first and u = cos^-1(x/2) for the second, but these do not seem to yield happy integrals. The first yields integral of 4sin^2(u) du (which is doable but not especially neat using a cos(2u) substitution), and the second yields integral of -tan^2(u) du, which I do not know how to do (but maybe I should?). Are there better substitutions than these?
    Your substitutions are spot on, couldn't be better. For the second one, notice that \tan^2u = \sec^2u-1, and the integral of  \sec^2u is \tan u.
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