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Math Help - Derivatives and Integrals with logarithmic functions

  1. #1
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    Derivatives and Integrals with logarithmic functions

    We went over this stuff in class, and I was fine doing the examples, but now I can't do the homework.
    The first is:
    Integral of (sec(x)dx)/(sqrt(secx+tanx))
    Any hints or ideas of where I can start?

    Also:

    Find the derivative, logarithmically, of y = (x*sin(x))/sqrt(sec(x))
    I tried this one just using the quotient rule for derivatives, and it turned out pretty bad.
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  2. #2
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    Quote Originally Posted by mistykz View Post
    Find the derivative, logarithmically, of y = (x*sin(x))/sqrt(sec(x))
    I tried this one just using the quotient rule for derivatives, and it turned out pretty bad.
    Take the hint:

    lny=lnx+ln(sinx)+0.5ln(cosx)

    and differentiate.....
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  3. #3
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    Hello, mistykz!

    What you covered in class may not have prepared you for this integral.


    \int \frac{\sec x\,dx}{\sqrt{\sec x + \tan x}}
    Multiply by \frac{\sec x + \tan x}{\sec x + \tan x}

    \int \frac{\sec x}{(\sec x + \tan x)^{\frac{1}{2}}} \cdot\frac{\sec x + \tan x}{\sec x + \tan x}\cdot dx \;=\;\int\frac{\sec x\tan x + \sec^2\!x}{(\sec x + \tan x)^{\frac{3}{2}}} \,dx


    Now let u \:=\:\sec x + \tan x




    Use logarithmic differentiation: . y \:= \:\frac{x\sin x}{\sqrt{\sec x}}
    If you differentiated logarithmically, there are no quotients involved!


    Take logs: .  \ln y \;=\;\ln\left(\frac{x\sin x}{(\sec x)^{\frac{1}{2}}}\right) \;=\;\ln(x) + \ln(\sin x) - \frac{1}{2}\ln(\sec x)


    Differentiate implicitly: . \frac{1}{y}\cdot y' \;=\;\frac{1}{x} + \frac{\cos x}{\sin x} -\frac{1}{2}\frac{\sec x\tan x}{\sec x} \;=\;\frac{1}{x} + \cot x - \frac{1}{2}\tan x


    Then: . y' \;=\;y\left(\frac{1}{x} + \cot x - \frac{1}{2}\tan x\right) \;=\;\frac{x\sin x}{\sqrt{\sec x}}\left(\frac{1}{x} + \cot x - \frac{1}{2}\tan x\right)

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