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Ignoring positive/negative values with trig substitutions?

  1. Sep 1, 2011 #1
    If I wanted to integrate [itex]\int \sqrt{1+x^2} dx[/itex], I would let [itex]x=\tan\theta[/itex] , which implies [itex]dx=\sec^2 \theta dx[/itex] so that I would have:

    [itex]\int \sqrt{1+x^2} dx = \int \sqrt{1 + \tan^2 \theta} \sec \theta d \theta = \int \sqrt{\sec^2 \theta}\sec^2\theta d\theta = \int \sec^3 \theta d \theta[/itex]

    It is this last equality that I am questioning. Why is it not [itex]\int |\sec \theta| \sec^2 \theta d \theta[/itex]?
  2. jcsd
  3. Sep 1, 2011 #2


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    Any value of x can be gotten by that substitution with -π/2 < θ < π/2. sec(θ) is positive in that domain.
  4. Sep 2, 2011 #3
    Thanks, that does clear it up a lot. So for all trig substitutions, then, there is an implied statement about the range of values which x can take and the corresponding range of values provided by the substitution?
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