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Math Help - integrate with trig functions

  1. #1
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    integrate with trig functions

    integrate:

    (1 - tan x) / (sec x)^2 dx

    i have tried substitutions and used trig identies but cannot seem to figure this out
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  2. #2
    o_O
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    Split the fraction into two and simplify each part into terms of \sin x and \cos x. So:

    \begin{aligned} \int \frac{1 - \tan x}{\sec^2 x} \ dx & = \int \left(\frac{1}{\sec^2 x} - \frac{\tan x}{\sec^2 x}\right)dx \\ & = \int \left( \cos^2 x - \sin x \cos x \right) dx \end{aligned}

    Now note that:
    • \cos^2 x = \frac{1 + \cos 2x}{2}
    • \sin x \cos x = \tfrac{1}{2}\sin (2x)
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