How prove this integral equation $\int_{a}^{b}\frac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\frac{1}{\sqrt{|f(x)|}}dx$ let $a>b>c>d$,and
$$f(x)=(x-a)(x-b)(x-c)(x-d)$$
show that
$$\int_{a}^{b}\dfrac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\dfrac{1}{\sqrt{|f(x)|}}dx$$
my try: maybe let
$$u=x+( )$$
such when $x=a,b$ then $u=c,d$?
if $$a+d=b+c$$
then we take
$$y=a+d-x$$
then we have $$\int_{a}^{b}\dfrac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\dfrac{1}{\sqrt{|f(x)|}}dx$$
other case,
 I can't,Thank you
 A: You need a fractional linear (a.k.a. projective linear) change of variable
$$
x' = \frac{rx+s}{tx+u},
$$
with coefficients $r,s,t,u$ chosen so that the map $x \mapsto x'$
is an involution taking $a,b,c,d$ to $c,d,a,b$.  Explicitly you can take
$$
r = ac-bd, \quad
s = bcd - acd + abd - abc, \quad
t = a - b + c - d, \quad
u = bd-ac
$$
(note that $u = -r$, which is necessary and sufficient for a
fractional linear transformation to be its own functional inverse).
This transformation takes each of the integrals
$\int_a^b dx \, / \sqrt{|f(x)|}$ and $\int_c^d dx \, / \sqrt{|f(x)|}$ 
to the other, proving that they are equal.
A: The general integral represents the Schwartz-Christoffel transformation of the upper halfplane to a rectangle, and the two integrals represent the differences between the images of the points $a$ and $b$ and $c$ and $d$ respectively. Since the opposite sides of a rectangle are equal, the two integral is the same (I am not sure why @Braindead, who is clearly aware of this fact, post this as an answer, but someone should :)
A: NOTE: I read the inequalities backwards, as $a<b<c<d$. Maybe that's what you meant too?
So here are some "elementary" methods that don't involve (too much) complex analysis.  My naive suspicion is that a simple linear transformation will not suffice, and that you actually need to use fractional linear transformation:


*

*Using a fractional linear transformation, you can actually map the interval $(a,b)$ to $(c,d)$ and $(c,d)$ to $(a,b)$ simultaneously with the right orientation. If you imagine that the real number line is joined as a circle with the point at infinity, it's sort of like a 'rotation' (with some distortion, of course.)

*You can map $a,b,c,d$ into $a',b',c',d'$ so that $b'-a' = d'-c'$, which is just another form of your equation $a'+d' = b'+c'$.
I personally like approach 2, but it all depends on what level of prior knowledge you have, and what sort of techniques you are willing / allowed to use.
I'll leave the detailed computation to you, but the motivation for using this method and also just "knowing" why the equation holds is essentially complex analysis (see @IgorRivin's answer). 
Note2: The first method requires you to verify that the particular linear fractional transformation keeps the integrand invariant. So you probably need to explicitly compute the coefficients.
Note3: The second method has the advantage that you don't actually need to explicitly compute the linear fractional transformation. Both integrals $\int_a^b$ and $\int_c^d$ will transform into $\int_{a'}^{b'}$ and $\int_{c'}^{d'}$ with a common integrand.  You still need to verify a few things though. Again, I'll let you think about it.
