Properties

1.  ∫^b_af ( x ) dx = -∫_b^a f ( x ) dx .  We can interchange the limits on any definite integral, all that we have to do is tack a minus sign onto the integral while we do.

2.  ∫_a^a f(x)dx = 0 . If the upper & lower limits are the similar then there is no work to accomplish, the integral is zero.

3.  ∫^b_a cf ( x ) dx = c∫^b_a f (x ) dx  , where c refer for any number.  Therefore, as with limits, derivatives, & indefinite integrals we can factor out a constant.

4.  ∫^b_a f ( x ) dx± g ( x ) dx = ∫^b_a f ( x ) dx± ∫^b_a g ( x ) dx .We can break up definite integrals across a sum or difference.

5.  ∫^b_a f ( x ) dx =∫^c_a f (x ) dx  +∫^b_a f (x ) dx  where c refer to any number.  This property is more significant than we might realize at first. One of the main utilizations of this property is to tell us how we can integrate function over the adjacent intervals, [a,c] and [c,b]. However note that c doesn't have to be between a & b.

6.  ∫^b_a f ( x ) dx =∫^b_a f ( t ) dt .The point of this property is to notice that as far as the function & limits are the similar the variable of integration that we utilizes in the definite integral won't affect the answer.

7. ∫_a^b c dx = c (b - a ) , c is refer for any number.

8.  If f ( x ) ≥ 0 for a ≤ x ≤ b then  ∫_a^b f(x) dx ≥ 0 .

9.  If f ( x ) ≥ g (x ) for a ≤ x ≤ b then  ∫_a^b f(x) dx ≥∫_a^b g(x) dx

10. If m ≤ f ( x ) ≤ M for a ≤ x ≤ b then m (b - a ) ≤ ∫_a^b f(x) dx ≤ M (b - a ) .

11. |∫_a^b f ( x ) dx|  ≤ ∫_a^b f ( x ) dx