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(Basic) Please give me advice on my u-substitution technique

  1. Nov 22, 2012 #1

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    1. The problem statement, all variables and given/known data

    I learned integration, but I think somewhere my technique is flawed, because I consistently get certain questions right and certain questions wrong. I don't know why I started doing it this way, but I always find DX by dividing it by DU and substitute it back in to the equation. Where can I improve and what am I doing wrong?

    2. Relevant equations

    Here are two equations. First one I get right, second one wrong.
    1) integral of (1 - 2x)^4 2dx
    2) integral of ((5)(3t - 6)^2

    3. The attempt at a solution

    1)
    U = 1 - 2x
    DU = -2 dx
    DX = DU/-2

    (U)^4 (2)(DU/-2)
    U^4 (-1)
    -1 integral of (1/5) U^5 + C
    = -1/5 (1-2x)^5 + c
    RIGHT


    2)
    U = 3t - 6
    DU = 3 dx
    dx = (du/3)

    5(u)^-2 du/3

    1/3 5(u)^-1
    (1/3) (-5/1) (3t - 6)^-1
    WRONG


    ?
     
    Last edited: Nov 22, 2012
  2. jcsd
  3. Nov 22, 2012 #2

    Simon Bridge

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    ... too many brackets. From your later statements I'm guessing you want:

    $$\int 5(3t-6)^{-2} dt$$

    However: Your substitution seems fine - I think you need to check how to integrate negative powers.
     
  4. Nov 22, 2012 #3

    SteamKing

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    Is the exponent 2 or -2?
     
  5. Nov 22, 2012 #4

    Simon Bridge

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    Wait:
    ... looks here like the power has been integrated properly (if it is -2). There is a minus sign missing in the previous line.
     
  6. Nov 22, 2012 #5
    I think that u-substitution is a technique that is used at times when it really isn't necessary. IMO, u-substitution is a good way to program a computer to do an integration, but it is usually not a good way to integrate (well, really we do u-substitution to find an anti-derivative, which we use to do integration.) So, how would I do this problem?

    Let's try to find the anti-derivative of[itex]2(1-2x)^4[/itex]. Now, we have something raised to the power of 4, and so a good guess at an anti-derivative would be to just write down:
    [itex](1/5)(1-2x)^5[/itex]. Now, we want to see if this is correct (ie is the derivative of our first guess what we started out with.) Well, the derivative is [itex]-2(1-2x)^4[/itex]. This is VERY close to what we started out with, in fact, it is a multiplicative constant away (what constant?) So, just divide our initial guess by this multiplicative constant and check it again, and we see that this is the correct anti-derivative.

    Now, if you try to do it like this, after a while, it will become second nature. It might have seen confusing the way I described it, but I think if you write it out, it will make sense. The fact of the matter is that u-substitution can sometimes make things MUCH simpler, but, IMO, these cases are few and far between. So, use u-sub when you HAVE to, but it should be a last resort (and after practice, you will realise which integrals really "need" u-sub.)
     
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