Properties of Definite Integrals: Definite Integral Definition
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We will be exploring some of the important properties of definite integrals and their proofs in this article to get a better understanding. Integration is the estimation of an integral. It is just the opposite process of differentiation. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. There are two types of Integrals namely, definite integral and indefinite integral. Here, we will learn about definite integrals and its properties, which will help to solve integration problems based on them.
Table of Contents:
Definite Integral Definition
Properties
Proofs
Example
Definite Integral Definition
If an integral has upper and lower limits, it is called a Definite Integral. There are many definite integral formulas and properties. Definite Integral is the difference between the values of the integral at the specified upper and lower limit of the independent variable. It is represented as;
Definite Integral Properties
Following is the list of definite integrals in the tabular form which is easy to read and understand.
Properties | Description |
Property 1 | |
Property 2 | Also |
Property 3 | |
Property 4 | |
Property 5 | |
Property 6 | |
Property 7 | 2 parts |
Property 8 | 2 parts or it’s an even function or it’s an odd function |
Properties of Definite Integrals Proofs
Property 1:
This is the simplest property as only a is to be substituted by t, and the desired result is obtained.
Property 2: , also
Suppose
If f’ is the anti-derivative of then use the second fundamental theorem of calculus, to get Also, if then
Hence, .
Property 3: If f’ is the anti-derivative of f, then use the second fundamental theorem of calculus, to get;
Let’s add equations (2) and (3), to get
Property 4:
Let, so that
Also, note that when and when . So, wil be replaced by when we replace a by t. Therefore,
from equation (4)
From property 2, we know that . Use this property, to get
Now use property 1 to get
Property 5:
Let, so that
Also, observe that when and when So, will be replaced by when we replace a by t. Therefore,
from equation (5)
From Property 2, we know that Using this property , we get
Next, using Property 1, we get
Property 6:
From property 3, we know that
Therefore,
Where, and
Let, so that
Also, note that when , and when Hence, when we replace a by t. Therefore,
from equation (7)
From Property 2, we know that . Using this property, we get
Next, using Property 1, we get
Replacing the value of I2 in equation (6), we get
Property 7:
we know that
Now, if then equation (8) becomes
And, if then equation (8) becomes
Property 8: or it is an even function and or it is an odd function.
Using Property 3, we have
Where,
Consider
Let,
Also, observe that when . Hence, will be replaced by when we replace a by t. Therefore,
From Property 2, we know that, use this property to get,
Next, using Property 1, we get
Replacing the value of in equation (9), we get
Now, if ‘f’ is an even function, then Therefore, equation (11) becomes
And, if ‘f’ is an odd function, then Therefore, equation (11) becomes
Now, let us evaluate Definite Integral through a problem sum.
Example
Question: Evaluate
Solution: Observe that,
Hence, using Property 3, we can write
Solving the integrals, we get