Definite Integration

The definite integral ∫ᵇₐ f(x)dx

The definite integral ∫ᵇₐ f(x)dx is defined as the limit of Riemann sums. We divide the interval [a,b] into n equal parts of width h = (b-a)/n, and ta

If F(x) is an antiderivative

If F(x) is an antiderivative of f(x), then ∫ᵇₐ f(x)dx = F(b) - F(a) = [F(x)]ᵇₐ. This powerful theorem connects differentiation and integration. For ex

Eight key properties simplify definite integral

Eight key properties simplify definite integral evaluation: (1) ∫ₐₐ f(x)dx = 0, (2) ∫ᵇₐ f(x)dx = -∫ₐᵇ f(x)dx, (3) ∫ᵇₐ f(x)dx = ∫ᵇₐ f(t)dt (variable in

For even functions f(

For even functions f(-x) = f(x), ∫₋ₐₐ f(x)dx = 2∫₀ₐ f(x)dx. For odd functions f(-x) = -f(x), ∫₋ₐₐ f(x)dx = 0. Example: ∫₋π/4^π/4 x³sin⁴x dx = 0 (odd f

For integrals of the form ∫₀^π/2

For integrals of the form ∫₀^π/2 sinⁿx dx and ∫₀^π/2 cosⁿx dx: If n is odd: result = (n-1)/n × (n-3)/(n-2) × (n-5)/(n-4) × ... × 4/5 × 2/3. If n is ev