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

Examples:

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