Properties of the definite integral, The definite integral over interval of zero length, Reverse order of integration, The constant multiple rule, The integral of the sum or difference of two functions, Internal addition of the definite integral, The definite integral of an odd function, The definite integral of an even function, The definite integral of a nonnegative and nonpositive functions, Relationship of the definite integrals of two functions over the same interval of integration, Absolute integrability, The mean value theorem, The average value of a function over the given interval, The indefinite integral

By reversing the upper and lower limits of integration to b < a, that is, passing through the x-axis from a to b in opposite direction, each difference, x₁ - a, x₂ - x₁, . . . , b - x_n_{- 1} of Riemann sums becomes negative while function values, f (x_i') can stay unchanged, therefore

The definite integral changes sign if the limits of integration interchange.

3) The constant multiple rule

Suppose f (x) is integrable over the interval [a, b] and c is any real number, then c f (x) is integrable over [a, b] such that

If integrand is multiplied by a constant, we can factor out the constant.

4) The integral of the sum or difference of two functions

Suppose f(x) and g(x) are integrable over the interval [a, b] then

The integral of the sum or difference of two functions is the sum or difference of the integrals of each function

5) Internal addition of the definite integral

Suppose f (x) is integrable over the interval [a, b] and c is a point inside the interval, i.e., a < c < b then, the additive property of the definite integral holds

Let for example c lies outside the interval such that c < a < b, and assume f (x) is integrable over the interval [c, b] then,

Therefore, the above rule holds for any arrangement of three points, a, b and c.

6) The definite integral of an odd function

If the integrand is an odd function f (x) = - f (-x), the graph of which is symmetrical about the origin, with the interval of integration [- a, a], then

7) The definite integral of an even function

If the integrand is an even function f (x) = - f (x), the graph of which is symmetrical about the y-axis, with the interval of integration [- a, a], then

8) The definite integral of a nonnegative and nonpositive functions

Suppose f (x) is integrable over the interval [a, b] and f (x) > 0 but not identically equal to zero, then

If f (x) is integrable over the interval [a, b] and f (x) < 0 but not identically equal to zero, then

9) Relationship of the definite integrals of two functions over the same interval of integration

Suppose f (x) and g (x) are integrable over the interval [a, b] and f (x) > g (x) for all x in [a, b], then

10) Absolute integrability

Suppose f is integrable over the interval [a, b], it can be shown that | f (x)| is also integrable on [a, b]

Since, - | f (x)| < f (x) < | f (x)|, then

what means the same as the above.

11) The mean value theorem

Suppose f (x) is integrable over the interval [a, b] and m and M are minimal and maximal value of the function, that is m < f (x) < M for all x in [a, b], then

Geometric meaning of the above inequality is that the area under the graph of f (x) over the interval [a, b] is contained inside the rectangles with the same base (b - a) and of the heights m and M.

Since f (x) is continuous in the interval [a, b] it

takes at least one time each value between m and M

inside the interval.

Therefore, there is at least one value x inside the

interval [a, b] such that

meaning, there exists the rectangle with the base

(b - a) and a height f (x) whose area equals the area under the graph of f (x) over the interval [a, b].

12) The average value of a function over the given interval

For a continuous function f over an interval [a, b], the average value of f (x) is defined as

Thus, the average value of a function f (x) over an interval [a, b] is equal to some value of the function between its minimal and maximal value inside the interval, as shows the above figure.

The indefinite integral

Assume the lower limit a of an integral is fixed and the upper limit is changing. Then, starting from the fixed ordinate at a, also changes the area A under the curve as its endpoint ordinate moves parallel.

As we evaluate the integral over an interval whose

right-hand endpoint x changes, the actual value of the

area is a function of x usually denoted F(x), so that

The integral F(x) is called the indefinite integral of f(x).

Note that the variable of integration is replaced by the

letter t as x now denotes the upper limit.

Note also that, the value of the above integral will change whether lower or upper limit of integration changes.

So for example, we choose other fixed point a₁ instead of a, such that a₁ > a then

Thus, as the point a₁ can be placed anywhere inside the interval [a, x], the same function f (x) has infinite number of indefinite integrals that differ only in a constant value F₀ that correspond to any chosen area over the interval [a, a₁], as shows the above figure.

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