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Homework Help: Can someone tell me how to integrate this?

  1. Mar 7, 2007 #1
    1. The problem statement, all variables and given/known data

    [tex]\Delta s=\int^{t_f}_{t_0}\sqrt{v_0^2-2v_0\sin\theta gt+g^2t^2}~dt[/tex]

    Can someone tell me how to integrate this? I am just an high-school sutdent and have basic integration knowledge.

    Thanks in advance. :approve:

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 7, 2007 #2


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    The way to do any integral is transform it into some standard form that you know how to deal with.

    Do you know how to do integrals containing [tex]\sqrt{a^2 + x^2}[/tex]?

    If so, you can get it into that form by "completing the square":

    [tex]g^2t^2 - 2v_0 \sin\theta gt + v_0^2[/tex]
    = [tex](gt - v_0 \sin\theta)^2 + \dots[/tex]

    Then substitute [tex] u = gt - v_0 \sin\theta[/tex]
  4. Mar 7, 2007 #3
    I understand the completing the square, but I don't know how to solve these integrals. Is doing inverse substitution?
  5. Mar 7, 2007 #4
    Once you have done the completing the square, and used the substitution, then you could use integration by parts to finish off the integral.
  6. Mar 7, 2007 #5
    Are you sure it is integration by parts?
  7. Mar 7, 2007 #6


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    If you have done hyperbolic functions (similar to trig functions) and their inverses, the relevant formula is

    [tex]\int{\frac{1}{\sqrt{1+x^2}} = \sinh^{-1}x[/tex]

    and integration by parts should get you there.

    If you just want the answer (e.g. to use in a project on the motion of projectiles, or whatever) go to http://integrals.wolfram.com/index.jsp
  8. Mar 7, 2007 #7
    I have just studied this functions the last hours and the inverse integration by substitution. I am just an high-school student but I have a calculus book, from where I am studying, it gives a good preparation for IPHO. I am now going to see integration by parts and how to apply in this situation.

    Thank you very much Aleph for helping me.
    Last edited by a moderator: Mar 7, 2007
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