# Can someone tell me how to integrate this?

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## Homework Statement

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

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

## The Attempt at a Solution

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 $$\sqrt{a^2 + x^2}$$?

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

$$g^2t^2 - 2v_0 \sin\theta gt + v_0^2$$
= $$(gt - v_0 \sin\theta)^2 + \dots$$

Then substitute $$u = gt - v_0 \sin\theta$$

I understand the completing the square, but I don't know how to solve these integrals. Is doing inverse substitution?

Once you have done the completing the square, and used the substitution, then you could use integration by parts to finish off the integral.

Are you sure it is integration by parts?

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

$$\int{\frac{1}{\sqrt{1+x^2}} = \sinh^{-1}x$$

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

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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