Must-Know Results:

• integral(0 to pi/2) sin^n x/(sin^n x + cos^n x) dx = pi/4
• integral(0 to pi/2) ln(sin x) dx = integral(0 to pi/2) ln(cos x) dx = -(pi/2)*ln 2
• integral(0 to pi) x*f(sin x) dx = (pi/2)*integral(0 to pi) f(sin x) dx
• integral(0 to 1) x^(m-1)(1-x)^(n-1) dx = (m-1)!(n-1)!/(m+n-1)! for positive integers
• integral(0 to infinity) e^(-x^2) dx = sqrt(pi)/2 (Gaussian integral)
• integral(0 to pi/2) sin x dx = integral(0 to pi/2) cos x dx = 1
• integral(0 to 1) 1/(1+x^2) dx = pi/4