Before evaluating any definite integral, check:

1. Is the integrand even/odd on a symmetric interval? (Zero or double)
2. Can King's Rule simplify? ($\frac{f}{f+g}$ pattern?)
3. Is the integrand periodic? (Reduce to one period)
4. Is there |f(x)|? (Split at zeros)
5. Is it a piecewise function? (Split at transition points)
6. Can Wallis' formula apply? (Powers of sin/cos on [0,pi/2])

After identifying the approach:

• For King's Rule: add I + I and simplify
• For substitution: change limits to match new variable
• For by parts: evaluate boundary terms F(b) - F(a)
• For Riemann sums: identify f, replace r/n by x, 1/n by dx

Key Numerical Values:

• integral(0,pi/2) sinx dx = 1
• integral(0,1) $\frac{1}{1+x^2}$ dx = pi/4
• integral(0,1) lnx dx = -1
• integral(0,pi/2) ln(sinx) dx = -$\frac{pi}{2}$ln2
• integral(0,pi/2) sin^$\frac{n}{sin^n+cos^n}$ = pi/4
• n!/$n^n$ ~ $\frac{1}{e}$^n (Stirling)

Common Errors to Avoid:

• FTC across discontinuities
• Wrong Wallis factor (even/odd mix-up)
• Not changing limits in substitution
• f(b-x) instead of f(a+b-x) in King's Rule