Integration By Substitution It's Been A While

[Lucretius]

Homework Statement

It's been god knows how long since I've had to use integration by substitution. I've totally forgotten it. I am trying to integrate to solve for the value of an electric field at a given point. The integral I am trying to solve is:

(2kz/4(pi)(epsilon_0)*1/(z^2+x^2)^(3/2) dx.

I know the answer is (2kz/4(pi)(epsilon_0)*(x/[z^2(z^2+x^2)^(1/2)]

Homework Equations

I'm sure I have to set u=(z^2+x^2). This makes du = 2x.

The Attempt at a Solution

I'm confused as to what to do now. The equation I'm integrating doesn't have an x in it anywhere. I don't think I can say du/2x=dx because I will have x's and u's in the integral, which is no good. However, I can't just ignore it.

Also, how did that z^2 get in there on the bottom? z is a constant in this integration and since u = z^2+x^2, the z-term drops right out. I feel terribly lost.

 

[Dick]

That's really confusing. Why don't you just post the actual problem you are working on and how you tried to solve it?

 

[Lucretius]

Sure thing.

I'm trying to find the electric field at an arbitrary distance z above a straight line segment, where the arbitrary distance z is measured above one of the endpoints of the line segment.

Relevant Equations:

We are given that the electric field of a line charge is $$\frac{1}{4 \pi \epsilon_0} \int_P \frac{\lambda (R)}{r^2}dl$$.

Attempt At Solution.

A little element of the electric field is going to be pointed in two directions. One will be in the z-direction. The other will be in the direction parallel to the line. Using the geometry of the problem, I found that

$$dE=\frac{1}{4 \pi \epsilon_0}\frac{\lambda dx}{r^2}(cos(\theta) \textbf{z}+sin(\theta)\textbf{x})=\frac{\lambda}{4 \pi \epsilon_0}\frac{\lamda dx}{r^3}(z\textbf{z}+x\textbf{x})$$, where the bold indicates unit vectors.

I split this up into two integrals. This is where I have to integrate by parts, and where I get stuck. I have for instance, one integral which is $$\frac{1}{4 \pi \epsilon_0} \int_0^L \frac{2 \lambda z}{(z^2+x^2)^{3/2}}}dx$$.

I set my u = (z^2+x^2) and my du is then 2xdx. I am confused because there is no x in my numerator, and I can't just say du/2x=dx because then I am going to have both x's and u's in my equation when I integrate it.

 

[gabbagabbahey]

I split this up into two integrals. This is where I have to integrate by parts, and where I get stuck. I have for instance, one integral which is $$\frac{1}{4 \pi \epsilon_0} \int_0^L \frac{2 \lambda z}{(z^2+x^2)^{3/2}}}dx$$.

I set my u = (z^2+x^2) and my du is then 2xdx. I am confused because there is no x in my numerator, and I can't just say du/2x=dx because then I am going to have both x's and u's in my equation when I integrate it.

Well, if $u = (z^2+x^2)$, then $2x=2\sqrt{u-z^2}$ right?...but I don't think that makes the integral any easier!

Try the substitution $$u=\frac{x}{\sqrt{x^2+z^2}}$$ instead

 

[Lucretius]

Well, if $u = (z^2+x^2)$, then $2x=2\sqrt{u-z^2}$ right?...but I don't think that makes the integral any easier!

Try the substitution $$u=\frac{x}{\sqrt{x^2+z^2}}$$ instead

Ah, I got it, I think.

If I use the u-substitution you suggest, I get $$du=\frac{1}{\sqrt{x^2+z^2}}-\frac{x^2}{(x^2+z^2)^{3/2}}dx$$, which, getting a common denominator yields:

$$\frac{z^2}{(x^2+z^2)^{3/2}}$$.

So $$\frac{du}{z^2}=\frac{dx}{(x^2+z^2}dx$$

My integral is then just $$\frac{\lambda z}{4 \pi \epsilon_0}\int_a^b \frac{du}{z^2}$$ which, after re-substituting for u back into x's, and plugging in the bounds 0 and L, I get

$$\frac{1}{4 \pi \epsilon_0} \frac{\lambda z L}{z^2(z^2+x^2)^{3/2}}$$.

Thanks, though I still wonder how I would have thought of that particular u-substitution on my own!