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Homework Help: Antiderivatives (find f for )

  1. Mar 20, 2010 #1
    1. This antiderivative question asks: Find f.
    The equation reads: f prime of x = square root of x to the 5th power minus 4 divided by the fifth root of x. (see below)

    f´ (x) = √(x^5 ) - 4/√(5&x)

    The answer in the back of the book is f(x) = (2/7)X^(7/2) - 5X^(4/5) + C but I got stuck trying to work it out.


    2. Here are the relevant equations:
    If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is: F(x) + C

    Antidifferentiation formulars:

    Function = cf(x) Particular antiderivative = cF(x)
    Function = f(x) + g(x) Particular antiderivative = F(x) + G(x)
    Function = x^n (n not equal -1) Particular antiderivative (x^n+1)/n+1


    3. My attempt at a solution reads as follows:
    X to the fifth power raised to the half power minus 4 times 5X raised to the minus a half

    = 〖〖(X〗^(5))〗^(1/(2 ))-4(5X^((-1)/2))

     
    Last edited: Mar 20, 2010
  2. jcsd
  3. Mar 20, 2010 #2

    Dick

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    The fifth root of x in terms of power is (x)^(1/5). So you have (x^5)^(1/2)-4/(x)^(1/5). Do you remember your rules of exponents? Can you change that into the sum of two simple powers of x?
     
  4. Mar 20, 2010 #3
    EDIT: Removed at Dick's request.
     
    Last edited: Mar 20, 2010
  5. Mar 20, 2010 #4
    Thank you for your replies Dick and JohnJDC.

    Ok, I think I remember the rules of exponents, so here goes another attempt:

    Question: f prime x = sq.root of x to the 5th power - 4 divided by the 5th root of x

    Simplify first:

    = ((x^5)^ 1/2) - 4/ (x^1/5)

    = (x^5/2) - 4x^-1/5

    most general antiderivative:

    = [x^(5/2 + 1) divided by (5/2 + 1)] - [4x^(-1/5 + 1) divided by (-1/5 + 1) ]

    = [ (x ^ 7/2) divided by (7/2) ] - [ (4x^4/5) divided by (4/5) ] + C

    = (2/7x^7/2) - [(4x^4/5) divided by 4/5 ] + C

    = (2/7x^7/2) - (5/4 * 4x^4/5) + C

    = (2/7x^7/2) - (5x^4/5) + C
     
    Last edited: Mar 20, 2010
  6. Mar 20, 2010 #5
    You're there--you just need to simplify the term on the right. Consider this:
    3x/(3/5) = 3x(5/3) = 5x
     
    Last edited: Mar 20, 2010
  7. Mar 20, 2010 #6
    Thank you JOhnJDC for your help
     
  8. Mar 20, 2010 #7

    Dick

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    Thanks!
     
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