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Integration Problem

  1. Jan 26, 2015 #1
    1. The problem statement, all variables and given/known data
    the integral of x^3 (x^2 + 20)^1/2
    2. Relevant equations
    use u substitution

    3. The attempt at a solution
    I think I have finally figured the problem out, can you confirm if this is the correct answer please?

    u=x^2 +20 x= sqrt(u-20)
    du= 2x dx
    integral of x^3 * sqrt( u) du/2x
    cancel the x's and move the 1/2 in front of the integral
    plug in the sqrt(u-20) for x
    1/2 integral of (sqrt(u-20))^2 * sqrt(u) du
    1/2 integral of u-20 * sqrt(u) du
    now I distribute the sqrt(u) to the (u-20) and get
    1/2 integral of u^3/2 - 2u^1/2
    then I integrated getting
    1/2[2/5u^5/2 - 4/3u^3/2]
    finally getting 1/2[2/5(x^2+20)^5/2 - 4/3 (x^2+20)^3/2] + C
    Last edited: Jan 26, 2015
  2. jcsd
  3. Jan 26, 2015 #2


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    Please post your working for that attempt.
  4. Jan 26, 2015 #3
    Moderator's Note: There was a previous version that the OP edited when he finally figured out what to do.
  5. Jan 26, 2015 #4


    Staff: Mentor

    The first term is correct, but should be simplified. The second term's coefficient is off.

    You can check to see if your answer is correct by differentiating your answer. If the derivative equals the original integrand, then all is good.
  6. Jan 26, 2015 #5


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    I think u2=x2+20 is a little simpler.
  7. Jan 27, 2015 #6
    I realized that I dropped the 0 on 20 and put 2 on the 4th line from the bottom. So then I got 40/3 as coefficient instead of 4/3 so my new final answer after distributing the 1/2 is
    ((x^2+20)^5/2)/5 - (20(x^2+20)^3/2)/3 +C is it right to distribute the 1/2 in?
  8. Jan 27, 2015 #7

    Ray Vickson

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    What did you get when you differentiated your "answer"? Did you get your original integrand, or did you get something else? If you got your integrand, then your answer is correct (if you differentiated correctly); otherwise, it is incorrect (or else you differentiated incorrectly). Please report what you got.
  9. Jan 27, 2015 #8


    Staff: Mentor

    This is correct, now, but as Ray and I said, you should differentiate your answer - you don't need us to confirm your answer (provided that you can differentiate correctly). Also, you don't distribute the 1/2 - you distribute the 1/3.

    I would simplify the answer to make it clearer and cleaner by putting the constants at the front of the two terms, like so:
    (1/5)(x2 + 20)^(5/2) - (20/3)(x2 + 20)^(3/2) + C
    (1/5)(x2 + 20)5/2 - (20/3)(x2 + 20)3/2 + C
    Best (using LaTeX):
    ##\frac{1}{5}(x^2 + 20)^{5/2} - \frac{20}{3}(x^2 + 20)^{3/2} + C##

    We have a page on how to get started with LaTeX: https://www.physicsforums.com/help/latexhelp/
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