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Solve using the second funtamental theorem of calculus

  1. Oct 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve using the second fundamental theorem of calculus
    [tex]\int [/tex] from 0 to 2 of 2x^2 (√x^3 + 1) dx



    2. Relevant equations

    Using the second fundamental theorem of calculus

    [tex]\int [/tex] from a to b of f(t) dt = g(b) - g(a)


    3. The attempt at a solution


    [tex]\int [/tex] from 0 to 2 of 2x^2 (√x^3 + 1)dx

    = 2/3 multiplyed by [tex]\int [/tex] from 0 to 2 of (√x^3 + 1) (3x^2 dx)



    This problem is solved in my book however I dont understand why the book added 2/3 and at the end of the equation (3x^2 dx) , in replace of 2x^2 .
     
    Last edited: Oct 17, 2009
  2. jcsd
  3. Oct 17, 2009 #2
    Is this what you're asked for? [tex]\int [/tex]2x²(√ x³ + 1)dx

    When I'm faced with a problem like this I usually take a guess at the function, differentiate it, and see how I can change it to get the proper answer. For example, say your guess is (√ x³ + 1). Differentiate it and see what you get - it's not the right answer but see what you can change
     
  4. Oct 18, 2009 #3
    Jimmy84 -

    Hint: Firstly, can you see that the two expressions for the integral are the same? Next, what is the derivative of [itex]x^3[/itex]. How can that be used in the new form of the integral expressed by your book.
     
  5. Dec 12, 2009 #4
    yea I can see that the two expressions for the integral are the same.

    but I'm not sure at all where does the derivative of x^3 come from.
     
  6. Dec 13, 2009 #5

    HallsofIvy

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    Actually, what's important is the derivative of x^3+ 1.

    Substitute u= x^3+ 1.
     
  7. Dec 13, 2009 #6
    So it goes like this. u = x^3+ 1.

    By the way it shouldnt be u = (the square root of x³ + 1) instead ?



    Then dx is replaced by 3x^2 , and because of that 2/3 should be before the integral to keep the expresion the same in this way

    2/3 multiplyed by [tex]\int [/tex] from 0 to 2 of (√x^3 + 1) (3x^2 dx)
     
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