Monday, November 17, 2008

Properties of definite integrals

Properties of definite integrals

1. ∫ab f(x)dx = -∫ba f(x)dx

2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx

3. ∫0a f(x)dx = ∫0a f(a-x)dx

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

5. If f(-x) = -f(x) (means f is an odd function), then
-aa f(x)dx = 0

6. ∫0af(x)dx = ∫0af(a-x)dx and
abf(x)dx = ∫abf(a+b-x)dx

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

Due to the above relation

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

abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.

In particular

0nTf(x)dx = n∫0Tf(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)≤∫abf(x)dx≤M(b-a)

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