Definite Integral

The definite integral is an integral function that has a lower limit and an upper limit. It can be defined as the difference between the integrated values of a function obtained by substituting the upper limit value and the function value obtained by substituting the lower limit value. It has start and end values in the form of a closed interval. For example, for the definite integral function

∫_a^b f(x) dx

The start value is also known as a lower limit is ‘a’ and the end value is known as the upper limit is ‘b’. It can also be written as [a,b].

Types of definite integrals

1. Improper integral- Just as there are integrals with finite limits, there are also integrals that have no finite limit or infinite limit. Such integrals are called improper integrals. For example,
    

   ∫_a^∞ f(x) dx

   
   Here, the integral has ‘a’ as the lower limit. But the upper limit is not defined. Hence, the function is unbounded. The interval of the function can be written as [a, ∞).


2. Line integral- The integral of a function that is to be integrated along a curve is called a line integral. For example,
    

   _c∫ f(x) dx

   
   Here, ‘c’ represents the curve along which the function f(x) is to be integrated. The curve can have points or limits for the integration of a portion of the curve.


3. Double integral- Similar to an ordinary definite integral, the only difference in double integral is that it is integrated twice with the same or different lower and upper limits. For example,
    

   ∫_a^b ∫_x^y f(x) dx

   
   Here, the first integration is done for the limits [x,y]. After obtaining a resultant function from the same, the function is then integrated along with the limits [a,b] which gives the final integrated function.

*Note- The same procedure is applicable for multiple integrals as well. Multiple Integrals are used in concepts taught in high-level mathematics and chemistry.

Properties of definite integrals

1. This property is used for reversing the order of the limits of the given function. For example,
    

   ∫_a^b f(x) dx = - ∫_b^a f(X) dx

   
   Here, while changing the order of the limits we must not forget to put a negative sign in the resultant integral.


2. This property of definite integral is used to split the limits of the given function. For example,
    

   ∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx



3. This property is used to integrate a constant. For example,
    

   ∫_a^b p dx = p(b-a)



4. This property is used to integrate functions which have a constant affixed with it. For example,
    

   ∫_a^b c . f(x) dx = c ∫_a^b f(x) dx



5. This property is useful while changing the variable of the integral function. For example,
    

   ∫_a^b f(x) dx = ∫_a^b f(y) dy



6. For a function f(x) ≤ g(x) then
    

   ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx



7. Property for addition and subtraction- For two functions f(x) and g(x)
    

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

    

   ∫_a^b [ f(x) - g(x)]dx = ∫_a^b f(x) dx - ∫_a^b g(x) dx



8. Property for division
    

   ∫_a^b [ f(x) / g(x)]dx = [∫_a^b f(x)dx] / [∫_a^b g(x)dx]