The Art and Craft of Problem Solving

338 CHAPTER 9 CALCULUS

Symmetry and Transformations

Problem 3.1 .26 on page 73 asked for the evaluation of the preposterously nasty integral

r/^2 dx

Jo 1 + (tanx)V2·

We assume you have already solved this--on your own after studying the similar-but­

easier Example 3.1.7 or with the assistance of the hint online. If not, here is a brief

sketch of a solution.

Obviously, the v'2 exponent is a red herring, so we make it an arbitrary value a

and consider

dx

f(x) :

=

1 + (tanx)a

The values of f(x) range from 1 to 0 as x ranges from 0 to n/2, with f(n/4) = 1/2

right smack in the middle. This suggests that the graph y = f(x) is "symmetric" with

respect to the central point (n/4,1/2), which in turn suggests that we look at the

transformed image of f(x), namely

g(x) := f(n/2 -x).

Armed with trig knowledge [for example, tan(n/ 2 - x) = cotx = l/tanx] the problem

resolves quickly, for it is easy to check that f(x) + g(x) = 1 for all x. Since

rn/2 rn/2

J

o f(x)dx =

J

o g(x)dx,

we're done; the integral equals n / 4.

Why beat this easy problem to death? To remind you that symmetry comes in

many forms. For example, we can say that two points are symmetric with respect to

circle inversion (introduced on page 307). This is just as valid a symmetry as a mere

reflection or rotation. So let's extract the most from the solution to Problem 3.1 .26.

It worked because there was a transformation x � n /2 - x that was invariant with

respect to integration, and that allowed us to simplify the integrand in the problem.

The moral of the story: search for "symmetry," by looking for "natural" transfor­

mations and invariants. Here's a challenging example.

Example 9.3.10 (Putnam 1993) Show that

l-lO( x 2 _X )^2 jir( X2_x )^2 rhl( x (^2) -x )^2

-^100 x^3 - 3x+^1

dx+

Ibl x^3 - 3x+ 1

dx +

Jf&\ x^3 - 3x + 1

dx

is a rational number.

Partial Solution: We sketch the idea, leaving the details to you. This is a very

contrived problem; the limits of integration are not at all random. Call the square root

of the integrand (same for all three terms) f(x) = (x^2 - x)/(x^3 - 3x + 1). Can you find

a transformation that either leaves f(x)^2 alone, or changes it in an instructive way, and

that does something sensible to the limits of integration?