Integral Help: Solving for (z^2+x^2)^(-3/2) with Independent Variable z

[lavster]

hi, i was readin through a textbook on electrodynamics and this integral has stumped me - can someone please tell me how to solve it-thanks!

\int\frac{1}{(z^2+x^2)^{\frac{3}{2}}}dx

where z is an independent variable, which can be treated as a constant

 

[lavster]

thats meant to be the integral of 1 over (z^2+x^2)^(3/2) wrt x. don't no how to mak it bigger!

 

[HallsofIvy]

hi, i was readin through a textbook on electrodynamics and this integral has stumped me - can someone please tell me how to solve it-thanks!

\int\frac{1}{(z^2+x^2)^{\frac{3}{2}}}dx

where z is an independent variable, which can be treated as a constant

Use the fact that 1+ tan^2(\theta)= sec^2(\theta) so that \sqrt{1+ tan^2(\theta)}= sec(\theta) to get rid of the square root.

Specifically, let x= z tan(\theta) so that z^2+ x^2= z^2+ z^2tan^2(\theta)= z^2(1+ tan^2(\theta)) and then \sqrt{z^2+ x^2}= z sec(\theta).

Of course, you will also need to set dx= z sec^2(\theta)d\theta so the integral becomes
\int \frac{1}{z^3 sec^3theta)} (z sec^2(\theta)d\theta= \frac{1}{z^2}\int \frac{1}{sec(\theta)}d\theta

Now, since sec(\theta)= 1/cos(\theta), that integral becomes
\frac{1}{z^2}\int cos(\theta)d\theta=\frac{1}{z^2} sin(\theta)+ C.

Since tan(\theta)= x/z, you can imagine a right triangle with "opposite side" x and "near side" z. Then the hypotenuse has length \sqrt{x^2+ z^2}, sin(\theta)= "opposite side over hypotenuse" or x/\sqrt{x^2+ z^2} so the integral is
\frac{xz^2}{x^2+ z^2}+ C

 

[g_edgar]

<br /> \int \bigl(z^{2} + x^{2}\bigr)^{-3/2} d x = \frac{x}{\sqrt{z^{2} + x^{2}} z^{2}}+C<br />

 

[elect_eng]

hi, i was readin through a textbook on electrodynamics and this integral has stumped me - can someone please tell me how to solve it-thanks!

\int\frac{1}{(z^2+x^2)^{\frac{3}{2}}}dx

where z is an independent variable, which can be treated as a constant

The above answer (from HallsofIvy) is not correct which can be verified by taking the derivative and checking.

It's not clear if you want the answer or the method. The answer is

{{x}\over{z^2 \sqrt{z^2+x^2}}} + C

Knowing the answer, you can prove it by taking the derivative and verifying. Reversing the steps gives you a method to solve, although seeing that method beforehand would not be easy and there are more elegant methods I'm sure. Perhaps there is a simple mistake in the above method and it can be salvaged.

 

[elect_eng]

Perhaps there is a simple mistake in the above method and it can be salvaged.

Just to add a comment here. I did check using the substitution x=ztan(u) as suggested by HallsofIvy. This works and leads to the correct answer. Although I didn't check the above derivation, I have to assume that there is a simple mistake. The basic suggested approach appears to be a good one.