Fundamental Theorem: integral(a,b) f(x)dx = F(b) - F(a) where F' = f

Basic Properties:

• integral(a,b) f = -integral(b,a) f
• integral(a,b) f = integral(a,c) f + integral(c,b) f
• integral(a,b) f(x)dx = integral(a,b) f(t)dt
• integral(a,b) [af + bg] = aintegral f + bintegral g

Symmetry Properties:

• King's Rule: integral(a,b) f(x) = integral(a,b) f(a+b-x)
• Even on [-a,a]: integral(-a,a) f = 2*integral(0,a) f
• Odd on [-a,a]: integral(-a,a) f = 0
• Queen's Rule: integral(0,2a) f = integral(0,a) [f(x)+f(2a-x)]

Periodicity:

• integral(0,nT) f = n*integral(0,T) f
• integral(a,a+T) f = integral(0,T) f

Leibniz Rule: d/dx integral(g(x),h(x)) f(t)dt = f(h(x))h'(x) - f(g(x))g'(x)

Riemann Sum: lim$\frac{1}{n}$ sum(r=0,n-1) f$\frac{r}{n}$ = integral(0,1) f(x)dx

Wallis' Formula: $I_n$ = integral(0,pi/2) $sin^n$ x dx

• $I_n$ = [$\frac{n-1}{n}$]*I_(n-2)
• Even n: final factor pi/2. Odd n: final factor 1.
• $I_0$=pi/2, $I_1$=1, $I_2$=pi/4, $I_3$=2/3, $I_4$=3pi/16

Standard Results:

• integral(0,pi/2) sin^$\frac{n}{sin^n+cos^n}$ = pi/4
• integral(0,pi/2) ln(sinx) = integral(0,pi/2) ln(cosx) = -$\frac{pi}{2}$ln2
• integral(0,pi) x*f(sinx) = $\frac{pi}{2}$*integral(0,pi) f(sinx)
• integral(0,1) x^(m-1)(1-x)^(n-1) = (m-1)!(n-1)!/(m+n-1)!

Inequality: m <= f <= M on [a,b] => m(b-a) <= integral f <= M(b-a) Triangle Inequality: |integral f| <= integral |f|