Properties of Definite Integrals

Properties of Definite Integrals

(a)

(b)

(c) , where c is a point inside or outside the

interval [a, b]

(d)

(e)  according as f(x) is an even or an odd function.

(f) if f(2a – x) = f(x)

           = 0 if f(2a − x) = −f(x)

(g)

(h) , where f(x) is periodic with period T and n ∈ I.

(i) If f(a+x) = f(x) then  

(j) if f(x) is a periodic function with

period a that is f(a + x) = f(x)

(k) If f is continuous on [a, b] then the integral function g defined by

      is differentiable in [a, b] and g′(x) = f(x),  ∀ x ∈ [a, b]

(l) If f(x) ≤ ϕ(x),  ∀ x ∈ [a, b] then

(m) .

(n) If m is the least  value and M is the greatest value of the function on the interval [a, b] then  m (b-a) ≤ .

(o) If a function f is integrable  and non negative on [a, b] and there exists a
point c ∈ [a, b] of continuity of f. for which f(c) > 0. Then .

(p) Leibnitz’s Rule – If f is continuous on [a, b] and u(x) and v(x) are differentiable functions of x whose values lie in [a,b] then

.

(q) If f is continuous on [a, b] then there exists a number c in [a, b] at which

f(c) = is called the mean value of the function f(x) on
the interval [a, b].

(r) If f(x) ≥ 0 on the interval [a, b] then

.