Can someone tell me how to integrate this?

[Gyroscope]

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.

Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[AlephZero]

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

 

[Gyroscope]

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

 

[ppyadof]

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

 

[Gyroscope]

Are you sure it is integration by parts?

 

[AlephZero]

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

 

[Gyroscope]

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.