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Math Help - integral proof

  1. #1
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    integral proof

    Hi, I hope someone can help with this problem:

    Show that

    integral_{x}^{1} dt/ (1+ t^2)

    is equal to

    integral_{1}^{1/x} dt/ (1 + t^2) if x>0.

    Thanks
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  2. #2
    Senior Member ecMathGeek's Avatar
    Joined
    Mar 2007
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    Quote Originally Posted by Wintaker99 View Post
    Hi, I hope someone can help with this problem:

    Show that

    integral_{x}^{1} dt/ (1+ t^2)

    is equal to

    integral_{1}^{1/x} dt/ (1 + t^2) if x>0.

    Thanks
    I'm assuming that _{x}^{1} means that the limits of integration are from x to 1, and that _{1}^{1/x} means that the limits of integration are from 1 to 1/x. If this is correct, I will rewrite the problem using my own notation.

    INT{x,1} 1/(1 + t^2) dt = INT{1,1/x} 1/(1 + t^2) dt

    It can be shown that:
    INT 1/(1 + t^2) dt = arctan(t) + C

    (If you need me to show how I know this, I can.)

    Therefore, the above integrations become:
    arctan(t) from {x,1} = arctan(t) from {1,1/x}
    arctan(1) - arctan(x) = arctan(1/x) - arctan(1)
    pi/4 - arctan(x) = arctan(1/x) - pi/4

    Let arctan(x) = y --> tan(y) = x
    Let arctan(1/x) = z --> tan(z) = 1/x --> cot(z) = x
    Thus x = tan(y) = cot(z) --> y = pi/2 - z

    So the above problem becomes:
    pi/4 - y = z - pi/4
    pi/4 - (pi/2 - z) = z - pi/4
    -pi/4 + z = z - pi/4
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