Pattern 1: King's Rule (Every Year) integral$\frac{0 to pi}{2}$ f$\frac{sinx}{(f(sinx)}$+f(cosx)) = pi/4. The exponent, function type, and disguise vary, but the core pattern is the same.

Pattern 2: x*sin $\frac{x}{1+cos^2 x}$ on [0,pi] (Very Frequent) Apply King's to get pi*integral sin $\frac{x}{1+cos^2x}$ dx. Substitute u=cosx for arctan result. Answer: $pi^2$/4.

Pattern 3: Riemann Sum (Most Years) Standard forms: 1/n sum $\frac{1}{1+r/n}$ = ln2, sum $\frac{r}{n}$^k = $\frac{1}{k+1}$, Stirling-type (n!)^$\frac{1/n}{n}$ = 1/e.

Pattern 4: Leibniz Rule (Frequent) F(x) = integral(0 to $x^2$) f(t) dt, find F'(x). Or: solve integral equation by differentiating.

Pattern 5: |sin x| or |cos x| with Periodicity Count periods, compute one-period integral, multiply.

Pattern 6: Floor/Fractional Part integral [x] dx: split at integers. integral {x} dx: use period 1 and integral(0,1) {x}dx = 1/2.

Pattern 7: Area Between Curves Find intersection points, determine which curve is above, integrate difference.