Can someone tell me how to integrate this?

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  • #1

Gyroscope

Homework Statement



[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:


Homework Equations





The Attempt at a Solution

 
  • #2
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]
 
  • #3
I understand the completing the square, but I don't know how to solve these integrals. Is doing inverse substitution?
 
  • #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.
 
  • #5
Are you sure it is integration by parts?
 
  • #6
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
 
  • #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.
 
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