Evaluate the integral from 0 to pi/2 of dx / (1 + tan(x)^{sqrt(2)}).

Evaluate the integral from 0 to pi/2 of dx / (1 + tan(x)^{sqrt(2)}).

Approach: Substitute x by pi/2 minus x, add the new expression to the original, and notice that the exponent plays no part in what survives.

pi/4. Call the integral I and apply the symmetry substitution u = pi/2 - x, which sends every angle to its complementary angle. Since tan(pi/2 - u) = 1/tan(u), the integrand becomes 1/(1 + tan(u)^{-sqrt(2)}) = tan(u)^{sqrt(2)}/(1 + tan(u)^{sqrt(2)}), so I also equals the integral from 0 to pi/2 of that expression. Adding the two forms gives 2I = the integral from 0 to pi/2 of (1 + tan^{sqrt(2)})/(1 + tan^{sqrt(2)}) dx = the integral of 1 dx = pi/2, so I = pi/4. The exponent independence is the point: any fixed positive power gives the same answer, and the same trick works for any function f with f(x) f(pi/2 - x) = 1.

Follow-up: What is the integral from 0 to 1 of x^a (1-x)^b dx when a and b are swapped, and what symmetry does the beta function inherit from this?

Key concepts: symmetry substitution, definite integral, complementary angle, exponent independence.