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Math Help - integral with radical and trigonometry

  1. #1
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    integral with radical and trigonometry

    Is there anyone who could walk me through this practice problem?:

    \int{\sqrt{1 + \frac{cosx}{sinx}}}\,dx

    I really struggle with integrals involving radicals or trigonometry.
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  2. #2
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    Re: integral with radical and trigonometry

    Hey infraRed.

    There is probably a simpler way but one suggestion I have is to use complex numbers. cos(x) = [e^(ix) + e^(-ix)]/2 and sin(x) = [e^(ix) - e^(-ix)]/2i.

    Given that you probably haven't done a lot of stuff like that can you mention the identities and concepts that you have studied in the course that you are taking?
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  3. #3
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    Re: integral with radical and trigonometry

    My apologies, but I actually misread the problem. (cos(x)/sin(x)) is supposed to be (cos(x)/sin(x))^2. I think that makes the problem a lot more workable.
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  4. #4
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    Re: integral with radical and trigonometry

    Hello, infraRed!

    \displaystyle \int \sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2}\,dx

    Apply some trig identities . . .

    \sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2} \;=\; \sqrt{1 + \cot^2x} \;=\;\sqrt{\csc^2x} \;=\;\csc x


    So we have: . \displaystyle \int \csc x\,dx


    Got it?
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  5. #5
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    Re: integral with radical and trigonometry

    Quote Originally Posted by Soroban View Post
    Hello, infraRed!


    Apply some trig identities . . .

    \sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2} \;=\; \sqrt{1 + \cot^2x} \;=\;\sqrt{\csc^2x} \;=\;\csc x


    So we have: . \displaystyle \int \csc x\,dx


    Got it?
    Which of course then can be written as

    \displaystyle \begin{align*} \int{\csc{(x)}\,dx} &= \int{\frac{1}{\sin{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)}}{\sin^2{(x)}}\,dx} \\ &= -\int{\frac{-\sin{(x)}}{1 - \cos^2{(x)}}\,dx} \\ &= -\int{\frac{1}{1 - u^2}\,du} \textrm{ after making the substitution } u = \cos{(x)} \implies du = -\sin{(x)}\,dx \end{align*}

    I'm sure you can go from here
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    Re: integral with radical and trigonometry

    integral with radical and trigonometry-29-sep-13.png
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