King's Rule (integral(a,b) f(x)dx = integral(a,b) f(a+b-x)dx) is the most powerful property. Here's a systematic approach to recognizing and applying it.

Type 1: $\frac{f}{f+g}$ Pattern (Most Common) If I = integral(0,a) f$\frac{x}{(f(x)}$+f(a-x)) dx, then King gives I = integral f$\frac{a-x}{(f(a-x)}$+f(x)) dx. Adding: 2I = integral 1 dx = a, so I = a/2.

Examples: sin^$\frac{n}{sin^n+cos^n}$ on [0,pi/2], sqrt$\frac{tan}{(sqrt(tan)}$+sqrt(cot)), ln(sinx)/[ln(sinx)+ln(cosx)].

Type 2: x*f(sinx) Pattern on [0,pi] King replaces x by pi-x; sin(pi-x) = sin(x). So I = integral(0,pi)(pi-x)f(sinx) dx. Adding: 2I = piintegral f(sinx) dx.

Type 3: Self-Cancellation (I = -I) If King's Rule transforms f(x) into -f(x), then I + I = 0, so I = 0. Example: integral(0,pi/2)$\frac{sinx-cosx}{(1+sinx*cosx)}$ dx — King turns it negative.

Type 4: Logarithmic Integrals integral(0,pi/4) ln(1+tanx) dx: King replaces tanx by $\frac{1-tanx}{(1+tanx)}$. The sum simplifies to ln2.

Type 5: Combining with Other Properties After King's Rule, the simplified integral may need even/odd properties, periodicity, or direct evaluation. King reduces complexity; other properties finish the job.

Recognition Checklist:

• Limits [0,pi/2] with sin and cos? -> King swaps them
• Limits [0,pi] with x*f(sinx)? -> King removes the x
• f$\frac{x}{(f(x)}$+g(x)) where g = f(complement)? -> I = $\frac{b-a}{2}$
• Symmetric-looking integrand on symmetric limits? -> Try King's