How Do You Solve This Complex Electrodynamics Integral?

[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!

$$\int^{1}_{-1}\frac{dz}{(1-\frac{v^2}{c^2}+\frac{v^2}{c^2}z^2)^\frac{3}{2}}=[\frac{z}{\frac{v^3}{c^3}(\frac{c^2}{v^2}-1)(\sqrt{\frac{c^2}{v^2}-1+z^2})}]^{1}_{-1}$$

thanks

 

[arildno]

Suppose you manage to rewrite this into the form:
$$K\int_{-1}^{1}\frac{dz}{(a^{2}+z^{2})^{\frac{3}{2}}}$$
a and K constants you should determine

Now, let:
$$z=a*Sinh(y)$$
where Sinh(y) is the hyperbolic sine function.

Your (indefinite) integral will then transform into:
$$\int\frac{dz}{(a^{2}+z^{2})^{\frac{3}{2}}}=\int\frac{a*Cosh(y)dy}{a^{3}Cosh^{3}(y)}=\frac{1}{a^{2}}\int\frac{dy}{Cosh^{2}(y)}$$
where Cosh(y) is the hyperbolic cosine function.
From here, it should be easy to proceed..

 

[I like Serena]

See http://en.wikipedia.org/wiki/List_of_integrals_of_irrational_functions.
There's a formula there that solves your integral, no substitutions needed.