Indefinite Integrals

In Calculus, the two important processes are differentiation and integration. We know that differentiation is finding the derivative of a function, whereas integration is the inverse process of differentiation. Here, we are going to discuss the important component of integration called “integrals” here.

Suppose a function f is differentiable in an interval I, i.e., its derivative f’ exists at each point of I. In that case, a simple question arises: Can we determine the function for obtained f’ at each point? The functions that could have provided function as a derivative are called antiderivatives (or primitive). The formula that gives all these antiderivatives is called the indefinite integral of the function. And such a process of finding antiderivatives is called integration.

The integrals are generally classified into two types, namely:

  • Definite Integral
  • Indefinite Integral

Here, let us discuss one of the integral types called “Indefinite Integral” with definition and properties in detail.

Indefinite Integrals Definition

An integral which is not having any upper and lower limit is known as an indefinite integral. 

Mathematically, if F(x) is any anti-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted,

∫f(x) dx = F(x) + C

We mention below the following symbols/terms/phrases with their meanings in the table for better understanding.

Symbols/Terms/Phrases  Meaning 
∫ f(x) dx  Integral of f with respect to x 
f(x) in  ∫ f(x) dx  Integrand
x in ∫ f(x) dx Variable of integration
An integral of f A function F such that F′(x) = f (x)
Integration The process of finding the integral
Constant of Integration  Any real number C, considered as constant function

Anti derivatives or integrals of the functions are not unique. There exist infinitely many antiderivatives of each of certain functions, which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason, C is customarily referred to as an arbitrary constant. C is the parameter by which one gets different antiderivatives (or integrals) of the given function.

Get Indefinite Integral calculator here.

Indefinite Properties

Let us now look into some properties of indefinite integrals.

Property 1: The process of differentiation and integration are inverses of each other in the sense of the following results:

Indefinite integrals 1

And

Indefinite integrals 2

where C is any arbitrary constant.

Let us now prove this statement.

Proof: Consider a function f such that its anti-derivative is given by F, i.e.

Indefinite integrals 3

Then,

Indefinite integrals 4

On differentiating both the sides with respect to x we have,

Indefinite integrals 5

As we know, the derivative of any constant function is zero. Thus,

Indefinite integrals 6

The derivative of a function f in x is given as f’(x), so we get;

Indefinite integrals 7

Therefore, 

Indefinite integrals 8

Hence, proved. 

Property 2: Two indefinite integrals with the same derivative lead to the same family of curves, and so they are equivalent. 

Proof: Let f and g be two functions such that

Indefinite integrals 9

Now,

Indefinite integrals 10

where C is any real number.

From this equation, we can say that the family of the curves of [ ∫ f(x)dx + C3, C3 ∈ R] and [ ∫ g(x)dx + C2, C∈ R] are the same. 

Therefore, we cay say that, ∫ f(x)dx = ∫ g(x)dx

Property 3: The integral of the sum of two functions is equal to the sum of integrals of the given functions, i.e., 

Indefinite integrals 11

Proof: 

From the property 1 of integrals we have,

Indefinite integrals 12

Also, we can write;

Indefinite integrals 13

From (1) and (2),

Indefinite integrals 14

Hence proved.

Property 4: For any real value of p,

Indefinite integrals 15

Proof: From property 1 we can say that 

Indefinite integrals 16

Also,

Indefinite integrals 17

From property 2 we can say that

Indefinite integrals 18

Property 5:

For a finite number of functions f1, f2…. fn and the real numbers p1, p2…pn,

∫[p1f1(x) + p2f2(x)….+pnfn(x) ]dx = p1∫f1(x)dx +  p2∫f2(x)dx + ….. +  pn∫fn(x)dx

Indefinite Integral Formulas

The list of indefinite integral formulas are

  • ∫ 1 dx = x + C
  • ∫ a dx = ax + C
  • ∫ xn dx = ((xn+1)/(n+1)) + C ; n ≠ 1
  • ∫ sin x dx = – cos x + C
  • ∫ cos x dx = sin x + C
  • ∫ sec2x dx = tan x + C
  • ∫ cosec2x dx = -cot x + C
  • ∫ sec x tan x dx = sec x + C
  • ∫ cosec x cot x dx = -cosec x + C
  • ∫ (1/x) dx = ln |x| + C
  • ∫ ex dx = ex + C
  • ∫ ax dx = (ax/ln a) + C ; a > 0,  a ≠ 1

Indefinite Integrals Examples

Go through the following indefinite integral examples and solutions given below:

Example 1:

Evaluate the given indefinite integral problem: ∫6x5 -18x2 +7 dx

Solution:

Given,

∫6x5 -18x2 +7 dx

Integrate the given function, it becomes:

∫6x5 -18x2 +7 dx = 6(x6/6) – 18 (x3/3) + 7x + C

Note: Don’t forget to put the integration constant “C”

After simplification, we get the solution

Thus, ∫6x5 -18x2 +7 dx = x6-6x3+ 7x+ C

Example 2: 

Evaluate f(x), given that f ‘(x) = 6x8 -20x4 + x2 + 9

Solution:

Given,

f ‘(x) = 6x8 -20x4 + x2 + 9

We know that, the inverse process of differentiation is an integration.

Thus, f(x) = ∫f ‘(x) dx=∫[6x8 -20x4 + x2 + 9] dx

f(x) = (2/3)x9 – 4x5 +(1/3)x3 + 9x+ C

Indefinite Integral vs Definite Integral

An indefinite integral is a function that practices the antiderivative of another function. It can be visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to signify getting the antiderivative. The indefinite integral is similar to the definite integral, yet the two are not the same. The below figure shows the difference between definite and indefinite integral.

Definite and indefinite integrals

Learn how to evaluate Definite Integrals here. To learn more about indefinite integrals, download BYJU’S- The Learning App.

Frequently Asked Questions – FAQs

How do you find the indefinite integral?

The process of finding the indefinite integral of a function is also called integration or integrating f(x). This can be expressed as:
∫f(x)dx = F(x) + C, where C is any real number.
We generally use suitable formulas which help in getting the antiderivative of the given function.
The result of the indefinite integral is a function.

What does an indefinite integral represent?

The indefinite integral represents a family of functions whose derivatives are f.

Are indefinite integrals and Antiderivatives the same?

The indefinite integral is similar to the definite integral, yet the two are not the same. We can get from an indefinite integral is a function, whereas from definite integral is an actual number.

Do definite integrals have C?

No, only indefinite integrals contain a real number C in the process of integration.

What are the bounds of an indefinite integral?

There are no bounds for an indefinite integral.

What is the indefinite integral of 0?

The integral of 0 is C because the derivative of C (or any constant) is zero. Therefore, ∫0 dx = C.

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