Before evaluating any definite integral, check:
- Is the integrand even/odd on a symmetric interval? (Zero or double)
- Can King's Rule simplify? (f/(f+g) pattern?)
- Is the integrand periodic? (Reduce to one period)
- Is there |f(x)|? (Split at zeros)
- Is it a piecewise function? (Split at transition points)
- 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) 1/(1+x^2) dx = pi/4
- integral(0,1) lnx dx = -1
- integral(0,pi/2) ln(sinx) dx = -(pi/2)ln2
- integral(0,pi/2) sin^n/(sin^n+cos^n) = pi/4
- n!/n^n ~ (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