27 Definite Integrals - Revision points - 1

Properties of definite integrals

1. ∫_a^b f(x)dx = -∫_b^a f(x)dx



2. ∫_a^b f(x)dx = ∫_a^c f(x)dx + ∫_c^b f(x)dx


3. ∫₀^a f(x)dx = ∫₀^a f(a-x)dx

4. If f(-x) = f(x) (means f is an even function), then
∫_-a^a f(x)dx = 2∫₀^a f(x)dx

5. If f(-x) = -f(x) (means f is an odd function), then
∫_-a^a f(x)dx = 0

6. ∫₀^af(x)dx = ∫₀^af(a-x)dx and
∫_a^bf(x)dx = ∫_a^bf(a+b-x)dx

7. ∫₀^af(x)dx = ∫₀^a/2f(x)dx+∫₀^a/2f(a-x)dx

Due to the above relation

∫₀^af(x)dx = 0 if f(a-x) = -f(x)
∫₀^af(x)dx = 2∫₀^a/2f(x)dx if f(a-x) = f(x)

8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫_a^xf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].


9. If f(x0 is periodic with period T then

∫_a^bf(x)dx = ∫_a+nT^b+nTf(x)dx, where n is an integer.

In particular

∫₀^nTf(x)dx = n∫₀^Tf(x)dx






If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫_a^bf(x)dx≤M(b-a)