Help with E-field integral please.

[Fjolvar]

Hello, I need help solving an integral in the line charge equation for my electrodynamics class. I'm using Griffiths third edition and I understand where the components of the problem come from, but I can't seem to solve the integral, although I'm sure it's very basic. It's been awhile since I took calc 3 and I can't seem to find any similar examples online. I attached the solution to the problem and put a red box around the two integrals (Ez and Ex) that I need help understanding. Just the basic rules that were used to integrate and find the solution would be much appreciated. Thank you!

 

[Fjolvar]

Here it is written out..

Integral of 1/[(z^2+x^2)]^(3/2) dx. How do I integrate this?

 

[kreil]

$$\int \frac{dx}{(x^2+z^2)^{3/2}} = \frac{x}{z^2 \sqrt{x^2+z^2}} + C$$

 

[Fjolvar]

I know the solution, but I wanted to know how to get it. What rules did you use?

 

[╔(σ_σ)╝]

trig substitution x= ztan( theta).

 

[Fjolvar]

Hmm, is there a list of these identities I can refer to? I'm still not seeing it heh.

 

[Fjolvar]

Can anyone please elaborate on the steps taken to solve this integral?

 

[╔(σ_σ)╝]

If you do not know how to use trigonometric substitutions you can pretty much forget about getting an answer to your question. I suggest you review your calc text or use your best friend, google.

 

[Fjolvar]

Alright so I tried to relearn trig sub. and attempted the problem but I'm going wrong somewhere. I scanned my work and attached it. If someone can quickly review and perhaps point out my mistake it would really mean a lot. Thanks.

 

[Inferior89]

If $$x = z \tan \theta$$ then $$dx = z \sec^2 \theta \mathbf{d \theta}$$

 

[Fjolvar]

Okay I finally figured out how to get the answer. Sorry to keep this thread going! At least I feel caught up on trigonometric substitution now. Thanks ╔(σ_σ)╝ and Inferior89.

 

[Inferior89]

Integrate and substitute back from $$\theta$$
$$\sin ( \arctan (x/z) ) = \frac{x}{\sqrt{x^2 + z^2}}$$

 

[Fjolvar]

I have another question about this problem. In the solution above they calculate each component of the vector field Ez and Ex by using cos theta and sin theta in place of the directional r hat vector. This may be a stupid question, but how do you determine whether to use cosine or sine in relation to a vector component.. such as Ex, Ey, or Ez? I'm comparing to the general equation used to find an Electric field of a line charge, E = (1/4pi epsilon o) * (r hat * lambda/ r^2) where they replaced r hat by cos theta.. Thank you.

 

[Fjolvar]

Anyone?