## Properties of the Definite Integral

The following properties are easy to check:

Theorem. If f (x) and g(x) are defined and continuous on [a, b], except maybe at a finite number of points, then we have the following linearity principle for the integral:

(i)  f (x) + g(x) dx = f (x) dx + g(x) dx;
(ii)  f (x) dx =  f (x) dx, for any arbitrary number .

The next results are very useful in many problems.

Theorem. If f (x) is defined and continuous on [a, b], except maybe at a finite number of points, then we have

(i) f (x) dx = 0;
(ii) f (x) dx = f (x) dx + f (x) dx;
(iii) f (x) dx = - f (x) dx;
for any arbitrary numbers a and b, and any c [a, b].

The property (ii) can be easily illustrated by the following picture: Remark. It is easy to see from the definition of lower and upper sums that if f (x) is positive then f (x) dx 0. This implies the following

If f (x) g(x) for x [a, b]  f (x) dx  g(x) dx  .

Example. We have (x2 - 2x)dx = x2 dx - 2 x dx  .

We have seen previously that x2 dx = and x dx = .

Hence (x2 - 2x)dx = - 2 = - . f (x)dx = 2  , f (x)dx = - 1  ,

find f (x)dx.

F(x) = f (t)dt

is increasing on [a, b].  f (x)g(x)dx    f (x) dx .  g(x) dx  .

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Mohamed A. Khamsi
Helmut Knaust