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Integrals look easy but I'm still confused

  1. Nov 28, 2012 #1
    Integrals...look easy but I'm still confused :(

    1. The problem statement, all variables and given/known data
    evaluate the integral ∫(36/(2x+1)^3)dx


    2. Relevant equations
    dx^n/dx = nx^(n-1)


    3. The attempt at a solution
    ∫(36/(2x+1)^3)dx = 6ln[(2x+1)^3]/((2x + 1)^2) ( I know this is wrong, but why??)


    ∫(36/(2x+1)^3)dx = -9/[(2x+1)^2)

    I know the second one I did is right...but Why was the first one wrong??
     
  2. jcsd
  3. Nov 28, 2012 #2
    Re: Integrals...look easy but I'm still confused :(

    when I tried to differentiate 6ln[(2x+1)^3]/((2x + 1)^2) ...I get back to the same one though
     
  4. Nov 28, 2012 #3
    Re: Integrals...look easy but I'm still confused :(

    oh sorry, nevermind...I already found what I did wrong....
     
  5. Nov 28, 2012 #4
    Re: Integrals...look easy but I'm still confused :(

    You don't even need that natural log here, you just need a u substitution. If you were given this integral, I assume that you know what a u substitution is, but if you don't, tell me and I'll show you what's going on here.
    [tex]\int \frac{36}{{2x+1)^3}}dx[/tex]
    [tex]36\int \frac{dx}{(2x+1)^3}[/tex]
    [tex]u=2x+1, du=2 dx \to dx=du/2[/tex]
    [tex]36\int \frac{du}{2u^3}[/tex]
    [tex]18\int u^{-3}du[/tex]
    [tex]18u^{-2}/-2+C[/tex]
    [tex]-9u^{-2}+C[/tex]
    [tex]-\frac{9}{(2x+1)^2}+C[/tex]
     
  6. Nov 28, 2012 #5

    Mark44

    Staff: Mentor

    Re: Integrals...look easy but I'm still confused :(

    This is wrong because it is NOT true that
    $$ \int \frac{1}{f(x)}dx = ln|f(x)| + C$$

    The correct formula is
    $$ \int \frac{1}{x}dx = ln|x| + C$$

    Another way to write this is
    ## \int x^{-1}dx = ln|x| + C##

     
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