Statement: integral(a to b) f(x) dx = integral(a to b) f(a+b-x) dx

Proof: Substitute u = a+b-x. When x=a, u=b; when x=b, u=a. dx = -du. The integral flips limits and the negative sign flips them back.

The f(x)/(f(x)+f(a-x)) Pattern: Let I = integral(0 to a) f(x)/(f(x)+f(a-x)) dx. By King's Rule: I = integral(0 to a) f(a-x)/(f(a-x)+f(x)) dx. Adding: 2I = integral(0 to a) 1 dx = a. Therefore I = a/2.

Examples:

• integral(0 to pi/2) sin^n x/(sin^n x + cos^n x) dx = pi/4 for any n
• integral(0 to 1) ln(1+x)/(2+x^2) type problems
• integral(0 to pi/2) sqrt(tan x)/(sqrt(tan x) + sqrt(cot x)) dx = pi/4