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Last Updated: Mar 27, 2024

Indefinite Integrals

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Introduction

If you have the basic knowledge of calculus, this blog will help you to learn one of the most important concepts of calculus, i.e. Indefinite Integrals

In this blog, you will learn about the different properties of Indefinite integrals with different examples. We have provided some examples at the end. It will help you understand the concept clearly and extend your knowledge of calculus. At the end of this blog, every concept will be clear to you. 

So, without further ado, let’s get started.

Definition

The integrals that do not have any upper and lower limits are the Indefinite Integrals.
Mathematically saying, given a function f(x), the anti-derivative of f(x) is ant function F(x) such that  f(x) = F’(x).
If F(x) is any anti-derivative of f(x) then the most general anti-derivative of f(x) is called an indefinite integral.

It is denoted by,

Where c is an arbitrary constant.

The value of the integrals of the functions is not unique. There can be infinitely many values depending on the values of the constant c from the set of the real numbers.

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Properties

The properties of the indefinite integrals are as follows - 

  1. The process of differentiation and integration are inverses of each other. That is,

    where c is an arbitrary constant.
    Proof:  

     
  2. For any real value A,

    Proof: 
  3. For any two functions f(x) and g(x),

    Proof:

     
  4. For a finite number of functions f1,f2,f3,........fn and for some real numbers A1, A2, A3……An,

Fundamental Integrals

Some of the fundamental integral formulas are shown here,

Examples

  1. Evaluate the given Integral


    Sol: 
  2. Evaluate the given integral

    Sol: 
  3. Evaluate the integral - 

    Sol: 
  4. Evaluate the integral - 

    Sol: 
  5. Evaluate the integral - 

    Sol:


     

Methods Of Integration

The different methods of integration are discussed below:

Integration by Substitution

The method of integration by substitution reduces the given integral to a standard form by the proper substitution.

Mathematical Definition

Mathematically saying, If g(x) is a continuously differentiable function, then if you want to evaluate the integration in the form of,

 

We substitute,

Now, this reduces the integral in the form,

After the evaluation, we substitute back the value of t.
 

Example

1. Evaluate the given integral,

Solution: 

2. Evaluate the given integral,

Solution:

Integration by Parts

Here we will see the integration by parts method.

Theorem

If u and v are two functions of x, then

Here, u is regarded as the first function and v is regarded as the second function of x.

Now the problem is how to choose the first function, right?
To choose the first function, you have just to follow a simple technique that is to choose the first function as the function which comes first in the word ILATE  where,

stands for Inverse trigonometric functions.

L stands for Logarithmic functions.

A stands for Algebraic functions.

T stands for Trigonometric functions.

E stands for Exponential functions.

Example

  1. Evaluate the given integral.

    Solution:
  2. Evaluate the given integral

    Solution: 

Integration by Partial Fractions

Let’s now discuss the method of integration by partial fractions.

Partial Fractions

Before learning about the Partial Fractions, let’s first see the Polynomial functions.

An expression of type

is called a polynomial of degree n.

Let’s now see what the Rational function is.

A function of the form P/Q where P and Q are polynomials is called the Rational function of x.
A rational function is called the Proper Rational function if the degree of the numerator is less than that of the denominator.

A rational function is called the Improper Rational function if the degree of the numerator is greater than that of the denominator.

If the rational function P(x) / Q(x) is Improper, then we divide P(x) by Q(x) so that it can be represented as the below form,
                        H(x) + (R(x) / Q(x))

Now, R(x) / Q(x) is a proper rational function.

 

Let’s now consider some cases related to partial fractions.

Cases of Partial Fractions

The different cases for the partial fractions are discussed below:

Case1: (The denominator can be expressed as a product of non-repeated linear factors)

Example

Evaluate the given integral

Solution:

 

Case2: (The denominator can be expressed as a product of linear factors such that some of them are repeating)

Example

Evaluate the given integral


Frequently Asked Questions

  1. What are indefinite Integrals?
    An Indefinite Integral is a function that takes the anti-derivative of another function. It is the reverse process of differentiation.
     
  2. What is the Indefinite Integral of 0?
    The Indefinite Integral of zero is constant.
  3. What is the meaning of the constant c in Indefinite Integrals?
    The constant C is used to include all the anti-derivatives of function f. There can be an infinite number of values of c  from the set of real numbers. Thus, for any function f, there can be an infinite number of anti-derivative possible.
     
  4. What is the main difference between Indefinite Integrals and Definite Integral?
    Definite integrals represent a number (when the lower and upper bound are constants), whereas the indefinite integrals represent a set of functions whose derivatives are the same.

Conclusion

In this article, we have extensively discussed the Indefinite Integrals topic and its implementation.

Here in this blog, we have covered every concept regarding indefinite integrals. We started with the basic introduction definition, moved to different properties, and solved some basic examples.

We hope that this blog has helped you enhance your knowledge regarding Indefinite Integrals and if you would like to learn more, check out our articles on ProbabilityPermutations and Combinations. Do upvote our blog to help other ninjas grow.

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Happy Reading!

Topics covered
1.
Introduction
2.
Definition
3.
Properties
4.
Fundamental Integrals
4.1.
Examples
5.
Methods Of Integration
5.1.
Integration by Substitution
5.1.1.
Mathematical Definition
5.1.2.
Example
5.2.
Integration by Parts
5.2.1.
Theorem
5.2.2.
Example
5.3.
Integration by Partial Fractions
5.3.1.
Partial Fractions
5.3.2.
Cases of Partial Fractions
5.3.3.
Case1: (The denominator can be expressed as a product of non-repeated linear factors)
5.3.4.
Example
5.3.5.
Case2: (The denominator can be expressed as a product of linear factors such that some of them are repeating)
5.3.6.
Example
6.
Frequently Asked Questions
7.
Conclusion