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Math Help - Trigonometric Integration (tangent^4)

  1. #1
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    Trigonometric Integration (tangent^4)

    Hello, the help is very appreciated.

    I'm asked to but not sure how to integrate the following:

    tan(x)^4*dx

    Many of our trig integration problems use the identities to simplify the integral, but I don't see how the identities could apply here.

    Wolfram Alpha suggested using a (foreign to our class, I believe) reduction formula in the link below.

    http://www.wolframalpha.com/input/?i=integrate+tan^4

    I don't think we used this formula in class though. Are there any other ways of solving this?
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  2. #2
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    Quote Originally Posted by NBrunk View Post
    Hello, the help is very appreciated.

    I'm asked to but not sure how to integrate the following:

    tan(x)^4*dx

    Many of our trig integration problems use the identities to simplify the integral, but I don't see how the identities could apply here.

    Wolfram Alpha suggested using a (foreign to our class, I believe) reduction formula in the link below.

    http://www.wolframalpha.com/input/?i=integrate+tan^4

    I don't think we used this formula in class though. Are there any other ways of solving this?
    Start by noting that \tan^4 (x) = \tan^2 (x) \cdot \tan^2 (x) = \tan^2 (x)(\sec^2 (x) - 1).
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  3. #3
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    I tried using the identity 1 + tan^2 = sec^2 and have gotten to

    -x + integral sec(x)^4 = -x + integral tan(x)^2 + integral tan(x)^2*sec(x)^2

    I tried then to continue with the same identity to get rid of the second term (with tan(x)^2), but it started flipping between integral of tan(x)^2 and sec(x)^2 indefinitely.
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  4. #4
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    Quote Originally Posted by NBrunk View Post
    I tried using the identity 1 + tan^2 = sec^2 and have gotten to

    -x + integral sec(x)^4 = -x + integral tan(x)^2 + integral tan(x)^2*sec(x)^2

    I tried then to continue with the same identity to get rid of the second term (with tan(x)^2), but it started flipping between integral of tan(x)^2 and sec(x)^2 indefinitely.
    For the first integral substitute for tan^2(x) again. In the second, substitute u = tan(x).

    Note: You're expected to know that \int \sec^2 (x) \, dx = \tan (x) + C.
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