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    How can determine whether the area given by the integral  \int_{0}^{\frac{\pi}{8}} \frac{dx}{\cos 3x} is positive or negative? I tried sketching the graph and think it's positive but is there an algebraic way to prove this without actually doing the integration?
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    You're pretty much there. As \sec 3x is always positive in [0, \pi/8], what can we therefore deduce about its integral over that interval?
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    (Original post by FXLander)
    How can determine whether the area given by the integral  \int_{0}^{\frac{\pi}{8}} \frac{dx}{\cos 3x} is positive or negative? I tried sketching the graph and think it's positive but is there an algebraic way to prove this without actually doing the integration?
    Show that \sec(3x)>0 in the interval [0,\frac{\pi}{8}]
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    (Original post by RDKGames)
    Show that \sec(3x)>0 in the interval [0,\frac{\pi}{8}]
    What's the best way to do so? I know that cos 3x > 0 in the interval [0, pi/8], so does that mean sec 3x>0 in that interval as well?
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    (Original post by FXLander)
    What's the best way to do so? I know that cos 3x > 0 in the interval [0, pi/8], so does that mean sec 3x>0 in that interval as well?
    Yes. Since \cos(3x)>0 in that interval, all you're then doing to those values is taking their reciprocals, so they stay positive.
 
 
 
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