## Properties of the Definite integral

• Area

b
∫ f(x) dx = (Area above x axis) − (Area below x axis)
a

• When the upper and lower limits are the same then the integral is zero.
a
∫ f(x) dx = 0
a

• The definite integral of 1 is equal to the length of the interval of integration:

b
∫ 1 dx = b -a
a

• Reversing the limits of integration changes the sign of the definite integral.

b                      b
∫ f(x) dx = – ∫ f(x) dx
a                      a

• The definite integral of the sum of two  functions is equal to the sum of the integrals of these functions:

b                                           b                b
∫ [f(x) dx + g(x) dx] = ∫ f(x) dx + ∫ g(x) dx
a                                          a                 a

• The definite integral of the difference of two functions is equal to the difference of the integrals of these functions:

b                                          b                b
∫ [f(x) dx – g(x) dx] = ∫ f(x) dx – ∫ g(x) dx
a                                         a                 a

• Suppose that a point c belongs to the interval [a , b]. Then the definite integral of a function
f(x)over the interval [a , b] is equal to the sum of the integrals over the intervals [a , c] and [c , b]:

b                    c                b
∫ f(x) dx = ∫ f(x) dx – ∫ f(x) dx
a                   a                 c

• The definite integral constant multiple

b                      b
∫c f(x) dx = c ∫ f(x) dx , c is any contant
a                      a

• The point of this property is to notice that as long as the function and limits are the same the variable of integration that we use in the definite integral won’t affect the answer.

b                      b
∫ f(x) dx = c ∫ f(t) dt ,
a                      a

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

b
∫ f(x) dx ≥ 0
a

Example.1 find out the integration of
∫ (4x2 – 2x) dx

Solution:

= ∫ 4x2 dx – ∫ 2x dx

= 4 ∫ x2 dx – 2 ∫ x dx

= 4x33 – 2 x22 + c

= 4x33 – x2 + c

= x2( 4x3 – 1) + c