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Math Help - another trig sub.

  1. #1
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    another trig sub.

    How would you do this
    integral of tan^4(x)sec^4(x)dx
    I know this breakups into integral of ((sec^2(x))^2 + integral of ((tan^2(x))^2
    I am not so sure where to go from here.
    Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by davecs77 View Post
    How would you do this
    integral of tan^4(x)sec^4(x)dx
    I know this breakups into integral of ((sec^2(x))^2 + integral of ((tan^2(x))^2
    I am not so sure where to go from here.
    Thanks.
    You know that (tan(x))^{\prime} = sec^2(x), so
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) sec^2(x) sec^2(x) dx

    What to do with the extra sec^2(x)? Well
    tan^2(x) + 1 = sec^2(x), so
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) sec^2(x) sec^2(x) dx = \int tan^4(x) (tan^2(x) + 1) sec^2(x) dx

    Now sub in y = tan(x) giving:
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) (tan^2(x) + 1) sec^2(x) dx  = \int y^4(y^2 + 1) dy

    You finish it from here.

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    You know that (tan(x))^{\prime} = sec^2(x), so
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) sec^2(x) sec^2(x) dx

    What to do with the extra sec^2(x)? Well
    tan^2(x) + 1 = sec^2(x), so
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) sec^2(x) sec^2(x) dx = \int tan^4(x) (tan^2(x) + 1) sec^2(x) dx

    Now sub in y = tan(x) giving:
    \int tan^4(x) sec^4(x) dx = \int tan^4(x) (tan^2(x) + 1) sec^2(x) dx = \int y^4(y^2 + 1) dy

    You finish it from here.

    -Dan
    Why wouldnt you do it my way..by breaking both tan^4(x) and sec^4(x)? Would that just make things harder?..so a good strategy is to break one piece of it up? Thank you!
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by davecs77 View Post
    Why wouldnt you do it my way..by breaking both tan^4(x) and sec^4(x)? Would that just make things harder?..so a good strategy is to break one piece of it up? Thank you!
    Because that's the way I'd do the integral and, honestly, I don't understand how
    \int tan^4(x) sec^4(x) dx = \int (sec^2(x))^2dx + \int (tan^2(x))^2 dx. It just doesn't look right to me, though I admit I haven't spent any time to disprove it.

    -Dan

    Edit: The two expressions tan^4(x) sec^4(x) and sec^4(x) + tan^4(x) aren't the same. Where did you get this from?
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  5. #5
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    Quote Originally Posted by topsquark View Post
    Because that's the way I'd do the integral and, honestly, I don't understand how
    \int tan^4(x) sec^4(x) dx = \int (sec^2(x))^2dx + \int (tan^2(x))^2 dx. It just doesn't look right to me, though I admit I haven't spent any time to disprove it.

    -Dan

    Edit: The two expressions tan^4(x) sec^4(x) and sec^4(x) + tan^4(x) aren't the same. Where did you get this from?

    OOPS i meant integral of (sec^2(x) - 1)^2 + integral of (tan^2(x) + 1)^2

    Thanks again..I was able to get the right answer now ;-)
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