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Math Help - Integration using trig substitution question

  1. #1
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    Integration using trig substitution question

    It's a common technique, when presented with (for example) \int \! \sqrt{u^2+1}\,\mathrm d u, to make a substitution such as u=\tan{\theta} and then we have that the previous integral is equal to \int \! \sqrt{\tan^2{\theta}+1}\sec^2{\theta}\, \mathrm d \theta=\int \! \sqrt{\sec^2{\theta}}\sec^2{\theta}\, \mathrm d \theta. Here is the point I'm concerned about - what allows us to proceed to \int \! \sec^3{\theta}\, \mathrm d \theta, as is the common practice? Our limits may not be such that sec{\theta}>0, so what justifies simplifying the square root portion?
    Thanks from alexmahone
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  2. #2
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    Re: Integration using trig substitution question

    I figured this out, this link supplies the details:
    http://ocw2.mit.edu/courses/mathemat...0_ChNnotes.pdf
    Thanks from alexmahone
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