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Simple Integrals

  1. Jul 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Question 1:
    Evaluate the indefinite integral.
    [tex]\int \frac{\cos x}{2 \sin x + 6} \, dx[/tex]

    Question 2:

    Evaluate the indefinite integral.
    [tex]\int \frac{2 \; dx}{x \ln (6 x)}[/tex]
    NOTE: The absolute value of x has to be entered as abs(x).

    3. The attempt at a solution
    Question 1:
    Let u = sinx, du = cosx dx
    = [tex]\int \frac{1}{2 u + 6} \, du[/tex]
    = [tex]\frac{1}{2} \int \frac{1}{u + 3} \, du[/tex]
    = [tex]\frac{1}{2} \int (u + 3)^{-1} \, du[/tex]
    = [tex]\frac{1}{2} * [ 1 + 1 ][/tex]
    = 1 + C

    Question 2:
    [tex]\int \frac{2 \; dx}{x \ln (6 x)}[/tex]
    Let u = ln(6x), du = 1 / 6x
    = [tex]12 \int \frac{1 \; du}{\ln (u)}[/tex]
    = [tex]12 \int \frac{1 \; du}{\ln (u)}[/tex]
    = [tex]12 \int (\ln (u))^{-1}\, du[/tex]
    Since inverse of ln is exp
    = [tex]12 \e^(u)[/tex]
    = [tex]12 \e^(ln(6x))[/tex]
    = 12 * 6 x = 72 x + C
     
  2. jcsd
  3. Jul 17, 2008 #2
    What is the Integral of [tex]\int\frac{dx}{x}[/tex]

    That's essentially what you have for # 1.

    You let [tex]u=\ln x[/tex]

    So what is it that you still have [tex]\ln x[/tex] in your Integral? Replace it with "u" completely.
     
  4. Jul 17, 2008 #3
    [tex]\int \frac{2 \; dx}{x \ln (6 x)}[/tex]
    Let u = ln(6x), du = 1 / 6x
    = [tex]12 \int \frac{1}{u} \ du[/tex]
    = [tex]12 [\ln(u)][/tex]
    = [tex]12 [\ln(ln(6x))][/tex] + C

    Are there any mistakes?
     
  5. Jul 17, 2008 #4
    Good! Did you get the first one too?
     
  6. Jul 17, 2008 #5
    Yep, I got Question #1. I made u = sinx+3 instead of u = sinx.

    Thank you rocomath!

    One more question, can i simplify ln(ln(x))?
     
  7. Jul 18, 2008 #6

    Defennder

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    Homework Helper

    I don't see how to simplify that any further.
     
  8. Jul 20, 2008 #7

    Gib Z

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    Homework Helper

    There indeed does seem to be mistakes. If you let u= ln (6x), make sure you use the chain rule to find du.
     
  9. Jul 20, 2008 #8

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Even simpler: ln(6x)= ln(x)+ ln(6) and the derivative of ln(6) is 0.
     
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