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Did I get the right anti derivate of this function?

  1. Mar 1, 2012 #1
    So, I have a question about finding the integral of x^3(1+x^2)^1/2

    This is basically what I did. Not sure if the answer is right or not.

    I did the following u substitution.

    u = 1 + x^2
    u - 1 = x^2

    du = 2xdx

    x^2du/2 = 2xdu * x^2

    x^2du/2 = x^3du

    (u - 1)du/2*u^1/2

    distributed the u

    1/2*u^3/2 - u^1/2
    u^2/2 = u

    1/2 ∫ u = 1/4*u^2

    = 1/4*(1+x^2)^2
  2. jcsd
  3. Mar 1, 2012 #2


    Staff: Mentor

    No. You can always check that your antiderivative (there's no such term as antiderivate) is correct, by differenting it. When you do this, you should get the original integrand.
  4. Mar 1, 2012 #3


    Staff: Mentor

    My first inclination, because of the square root of the sum of squares, would be a trig substitution. Another approach would be to use integration by parts.

    If an ordinary substitution works, then that would be preferred, if you can find a suitable one, although I'm not sure that this the way to go in this problem.

    Your work is hard to follow, so I didn't try to pick out where you went wrong. Make it easier to follow be starting with x3√(1 + x2) dx, and make you substitution in a systematic way.
  5. Mar 1, 2012 #4
    I was sure I got this right. Well this is a bummer.
  6. Mar 1, 2012 #5


    Staff: Mentor

    Actually, your substitution will work.

    I would write this as
    ∫(1/2)(u - 1)u1/2du
    = (1/2)∫(u3/2 - u1/2)du
    Now, carry out the integration and undo your substitution.
  7. Mar 2, 2012 #6
    Yeah, I made a small algebra mistake. Thought you could subtract the exponents but you can't. Only when you divide I suppose. My algebra roots aren't very strong. What do you recommend I do to strengthen my roots so it doesn't hurt me.
  8. Mar 2, 2012 #7


    Staff: Mentor

    If you still have your algebra textbook, spend a little time regularly brushing up on the areas where you feel you're weak. If you don't still have your textbook, take a look at khanacademy.com. They have many videos on all sorts of areas of mathematics.
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