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Math Help - Troubling Trig Integral

  1. #1
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    Troubling Trig Integral

    I am currently working on an integral that I finally found the answer to after approaching differently, but I would like to know why my first approach failed -- it led to undefined trig functions and/or ln 0. As far as I can tell, by math logic is okay, unless I'm missing something.

    <br />
\int_{0}^{\pi/2}\frac{cost}{1 + sint} dt <br />

    My initial approach was to multiply by \frac{1 - sin t}{1 - sin t} leading to \int\frac{cost(1 - sin t)}{1 - sin^2 t} and then to \int\frac{cost(1 - sin t)}{cos^2 t} and then to \int\frac{1 - sin t}{cos t}. Rewriting, \int_{0}^{\pi/2}\frac{1}{cos t} dt - \int_{0}^{\pi/2}tan t dt. This leads to \int_{0}^{\pi/2}sec t dt - \int_{0}^{\pi/2}tan t dt. But when I attempted to integrate, it failed. Simple u-subst. works u = 1 + sint,
    du = cost dt, but why does the other approach fail?
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  2. #2
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    In both the approachs you get the same answer.
    Intg(secθ) - Intg(tanθ)
    = ln(secθ + tanθ) -ln(secθ)
    = ln[(secθ + tanθ/secθ]
    = ln[ 1 + sinθ]
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  3. #3
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    My second step was: Intg tant dt = -ln|cost|. I failed to recognize the log property and rewrite -ln|cost| as ln|cost^-1| = ln|sect|.

    That was it. Thank you, it was driving me crazy!
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