1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral Evaluation

  1. Apr 24, 2012 #1
    1. The problem statement, all variables and given/known data

    This is part of a larger problem about finding the distance travelled by the particle over the interval 0≤t≤3. I need to solve the integral

    [itex]\int^3_0 \sqrt{t^4+t^2} \ dt[/itex]


    3. The attempt at a solution

    So, is it correct to rewrite [itex]\sqrt{t^4+t^2}[/itex] as [itex]t \sqrt{t^2+1}[/itex] and then use integration by parts?

    I'm confused because when I use Wolfarm online integrator to evaluate

    [itex]\int \ \sqrt{t^4+t^2} = \frac{(t^2+1) \sqrt{t^4+x^2}}{3t}[/itex]

    But when I use the other expression I get:

    [itex]\int \ t \sqrt{t^2+1} = \frac{1}{3} (t^2+1)^{3/2}[/itex]

    So which one is correct? :confused:
     
  2. jcsd
  3. Apr 24, 2012 #2
    Both seem to be valid. You had the right idea to calculate the integral by hand. Factor out the t^2, then remove it from the radical leaving [itex]t\sqrt{t^{2}+1}[/itex] and use integration by parts.
     
  4. Apr 24, 2012 #3

    Mark44

    Staff: Mentor

    Yes, it can be rewritten this way. I would use an ordinary substitution, though, not integration by parts.
    There's a typo above - you shouldn't have x in the antiderivative.
     
  5. Apr 25, 2012 #4
    That's a typo, I meant to say: [itex]\frac{(t^2+1)\sqrt{t^4+t^2}}{3t}[/itex]

    So, my question was: shouldn't the final intrgral be unique regardless of how I simplify the integrand? Because I got two clearly different answers. Which method was correct?
     
  6. Apr 25, 2012 #5

    sharks

    User Avatar
    Gold Member

    The final integral will depend on your integration technique, but should be unique after simplification.

    The second one is correct as it's in its most simplified form. Wolfram's format is computer-generated, so although different, it's equivalent.
     
  7. Apr 25, 2012 #6

    Mark44

    Staff: Mentor

    If you use two techniques to evaluate an indefinite integral (no limits of integration), you won't necessarily get the same things, but they can differ by at most a constant.

    If you use two techniques to evaluate a definite integral, they have to produce the same result (unless you make a mistake).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral Evaluation
  1. Evaluating integral (Replies: 7)

  2. Evaluate the Integral (Replies: 5)

  3. Evaluate this integral (Replies: 4)

  4. Evaluate the integral (Replies: 7)

  5. Integral evaluation (Replies: 5)

Loading...