# 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

ab 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,

abxy 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,

ab f(x) dx = - ∫ba 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,

ac f(x) dx = ∫ab f(x) dx + ∫bc f(x) dx

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

ab p dx = p(b-a)

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

ab c . f(x) dx = c ∫ab f(x) dx

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

ab f(x) dx = ∫ab f(y) dy

6. For a function f(x) ≤ g(x) then

ab f(x) dx ≤ ∫ab g(x) dx

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

ab [ f(x) + g(x)]dx = ∫ab f(x) dx + ∫ab g(x) dx

ab [ f(x) - g(x)]dx = ∫ab f(x) dx - ∫ab g(x) dx

8. Property for division

ab [ f(x) / g(x)]dx = [∫ab f(x)dx] / [∫ab g(x)dx]

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