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.

Definite Integral Properties

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