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Process for finding a definite integral

  1. Feb 28, 2013 #1
    1. The problem statement, all variables and given/known data

    given that 01 xndx = 1/ n+1
    for integers n [itex]\geq[/itex] 0
    calculate these integrals

    2. Relevant equations
    01 (1+x+x2) dx

    3. The attempt at a solution
    I have found the definite integral of 01 (1+x+x2) dx
    to = 1.8333.... (ie. 11/6)
    but all I did was use my TI-84 to find the definite integral. is it possible for me to find this solution by hand without a calculator?
     
  2. jcsd
  3. Feb 28, 2013 #2

    Fredrik

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    Yes, it is. You need to find a way to rewrite ##\int_0^1(1+x+x^2)\,\mathrm{d}x## so that you can use the formula you were given.
     
  4. Feb 28, 2013 #3
    could I use this?
    011dx + ∫01xdx + ∫01 x2dx

    where would I go from there if so?
     
  5. Feb 28, 2013 #4

    Fredrik

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    Yes, that's the correct first step. Now you just use the formula you were given on each of the three terms.
     
  6. Feb 28, 2013 #5
    I think I have it now thanks Fredrik!
    I could use 1/1 + 1/2 + 1/3 (as 1 = x0)
    Which would be 1 + 0.5 + 0.33 = 1.833

    is this the correct process for working out these types of problems, just so I know in future?
     
  7. Feb 28, 2013 #6

    Fredrik

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    Yes, it is. Your book should contain a theorem that says that for all integrable functions f,g, and all real numbers a,b, we have
    $$\int_a^b \left(f(x)+g(x)\right)\,\mathrm{d}x =\int_a^b f(x)\,\mathrm{d}x +\int_a^b g(x)\,\mathrm{d}x.$$ This allows you to deal with one term at a time.
     
  8. Mar 1, 2013 #7
    yes I see it now, along with a few similar theorems, it was located in a previous chapter of my textbook and I hadn't read it yet.
    Thanks again for pointing that out, and for your help.

    -5ym
     
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