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Find the integral of x/(x-6) with substitution

  1. Mar 6, 2010 #1
    [tex]\int\frac{x}{x-6}dx[/tex]


    [tex]u=x-6[/tex]

    [tex]\int \frac{u+6}{u}du[/tex]

    [tex]\int 1+\frac{6}{u} du[/tex]

    [tex]u+6ln|u|+C[/tex]

    [tex]x-6+6ln|x-6|+C[/tex]

    this appears to be incorrect although it seems logical, also if somone could please tell me the syntax for the definte integral

    also similar case here

    [tex]\int \frac{x^2}{x+4}dx[/tex]

    [tex]u=x+4[/tex]
    [tex]\int \frac{(u-4)^2}{u}du[/tex]
    [tex]\int \frac{u^2-8u+16}{u}du[/tex]
    [tex]\int u-8+\frac{16}{u} du[/tex]
    [tex]\frac{u^2}{2}-8u+16ln|u|+C[/tex]
    [tex]\frac{(x+4)^2}{2}-8(x+4)+16ln|x+4|+C[/tex]
     
    Last edited: Mar 6, 2010
  2. jcsd
  3. Mar 6, 2010 #2
    Re: integral

    When do you a u-sub don't forget to find du.

    Hint on why that sub won't work. d/du of x-6 isn't x.
     
  4. Mar 6, 2010 #3
    Re: integral

    in the first problem

    x=u+6 and du=dx
     
  5. Mar 6, 2010 #4
    Re: integral

    For the second problem, have you considered Polynomial long division, or do you have to use a substitution?
     
  6. Mar 6, 2010 #5
    Re: integral

    How'd you end up with the u+6 on the top?

    Ohh...I see what you did.

    The only thing I can think of where the mistake was made. I tried it using long division first and got the correct answer.

    Someone smarter than I will have to answer that as I don't solve integrals that way.
     
  7. Mar 6, 2010 #6
    Re: integral

    For the first, I think more algebra is required than substitutions, you just need to be cleaver.
     
  8. Mar 6, 2010 #7
    Re: integral

    If you are getting the same error in both of those, maybe your substitution trick is the problem.

    Is that even allowable?
     
  9. Mar 6, 2010 #8

    Char. Limit

    User Avatar
    Gold Member

    Re: integral

    Actually, it is allowable... look up integration by substitution on wikipedia.

    Your first answer is right. Just remember that since C is any constant number, the -6 is merged into C to give...

    [tex]x+6log(x-6)+C[/tex]
     
  10. Mar 6, 2010 #9

    Mark44

    Staff: Mentor

    Re: integral

    The first one can be done by polynomial long division or by substitution. I don't see anything wrong with your answer. If you think that your answer is incorrect, it might only be that your answer differs from the "correct" answer by a constant. If you take the derivative of your result, you get the original integrand. erok81 is not giving you good advice.

    For the second problem, one approach is to use polynomial long division, although your substitution works just fine. Again, the two different approaches give differnent appearing antiderivatives, but they differ only by a constant.
     
  11. Mar 6, 2010 #10

    Mark44

    Staff: Mentor

    Re: integral

    He didn't forget it; he just didn't explicitly show how he found du
    And that's not what he did. Why would you want to take the derivative with respect to u of x - 6? That doesn't make any sense in this problem.
     
  12. Mar 7, 2010 #11
    Re: integral

    There is more than one way to solve this problem. He thought his method was wrong, so I did it my way - which wasn't a u-sub to start with. So it wasn't bad advice, just a different way to solve it. Notice I did say someone smarter than I, like you, needs to come in and clarify since I wouldn't solve this problem the way he did. :)

    He did take the deritive of his u-sub and got dx. I commented on it and he explained.

    To make the last part clear so I make mistakes on my homework. If you have a problem say x/x^2+6. Your u-sub is u=x^2+6 then you take the deritive and get du = 2x dx or du/2 = x dx. What is wrong with my method? Sure it skips 2-3 steps, but it isn't wrong. Is it? I hope not since I've made it through calc I and most of II doing them that way.
     
  13. Mar 7, 2010 #12

    Mark44

    Staff: Mentor

    Re: integral

    No, that's fine. The bad advice was telling the OP why a substitution won't work (it does work), and telling him/her that d/du (x - 6) isn't x. Maybe you meant d/dx(x - 6).
     
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