Indefinite Integral Introduction

Differentiation and integration are two of the most crucial procedures in calculus to understand. It is well-known that differentiation is the process of determining the derivative of a function, whereas integration is the inverse of differentiation. The term “integrals” refers to a critical component of integration that we will address in this section.

For example, suppose that a function f differs between two points on an interval I, which means that its derivative f’ exists at each point on I. As a result, we are faced with a straightforward question: Can we establish the function for acquired f’ at each point? Antiderivatives are functions that could have delivered the same result as a derivative of another function (or primitive). The indefinite integral of the function is the formula that yields all of these antiderivatives and is used to represent them. Integration is the term used to describe the process of discovering antiderivatives.

The integrals are commonly divided into two categories, which are as follows:

• Indefinite Integral 
• Definite Integral

Let’s take a closer look at one of the integral types, the “Indefinite Integral,” and examine its definition and properties in greater depth.

Indefinite Integrals Definition

An indefinite integral is an integral that does not have any upper or lower bounds and therefore has no upper or lower bounds.

According to mathematics, if F(x) is an anti-derivative of f(x), then the greatest generic anti-derivative of f(x) is referred to as an indefinite integral and indicated by the symbol

∫f(x) dx = F(x) + C

Both the functions’ antiderivatives and integrals are equivalent to one another and cannot be distinguished from one another.

Every one of certain functions has an infinite number of antiderivatives, which may be derived by selecting C at random from the set of all possible real numbers.

As a direct consequence of this, C is frequently referred to as an arbitrary constant throughout the body of scientific literature. Different antiderivatives (or integrals) of the following function can be obtained by varying the value of the parameter C.

Characteristics of Indefinite Integrals

• Property 1: The processes of differentiation and integration are diametrically opposed to one another in the sense that the following outcomes are obtained:

d/dx∫f(x)dx =f(x)

And,

∫f'(x)dx =f(x)+C

In this case, C can be any arbitrary constant.

Let us now put this statement to the test.

To demonstrate this, consider a function f such that its anti-derivative is supplied by the function F.

d/dxF(x)dx =f(x)

Then,

∫f(x)dx =f(x)+C

We have the following results when we differentiate both sides with respect to x:

d/dx∫f(x)dx =d/dx(F(x)+C)

It is well-known that the derivative of a constant function is equal to zero. Thus,

d/dx∫f(x)dx =d/dx(F(x)+C)

                  = d/dxF(x)

                  =f(x)

The derivative of a function f in a variable x is denoted by the symbol f'(x), as follows:

f'(x)= d/dx f(x)

Therefore

∫f'(x)dx =f(x)+C

As a result, it has been demonstrated.

• Property 2: Two indefinite integrals with the same derivative lead to the same family of curves, and as a result, they are comparable.

Proof: Assume that f and g are two functions that are equivalent.

d/dx∫f(x)dx =d/dx∫g(x)dx 

Or,

d/dx[∫f(x)dx -∫g(x)dx ]=0

Now,

∫f(x)dx =∫g(x)dx +C

Or,

∫f(x)dx -∫g(x)dx =0

where C might be any positive or negative real number.

It follows from this equation that the families of curves of  [ ∫ f(x)dx + C₃ , C₃ ∈ R] and [ ∫ g(x)dx + C₂, C₂ ∈ R]  are the same.

So, we may claim that, in conclusion ∫f(x)dx =∫g(x)dx.

• Property 3: It can be shown that the integral of the sum of two functions is equivalent to the sum of the integrals of the two functions that are presented, i.e.,

∫[f(x)+g(x)]dx = ∫f(x)dx+∫g(x)dx

Proof:

We can deduce the following from the property 1 of integrals:

d/dx∫[f(x)+g(x)]dx = f(x)+g(x)….(1)

In addition, we can write;

d/dx[ ∫f(x)dx+∫g(x)dx] =d/dx∫f(x)dx+ d/dx∫g(x)dx = f(x)+g(x)…..(2)

based on numbers 1 and 2

∫[f(x)+g(x)]dx = ∫f(x)dx+∫g(x)dx

As a result, it was proven.

• Property 4 states that for any real value of p,

∫pf(x)dx = p∫f(x)dx

Proof: From the first  property, we may deduce that

d/dx∫pf(x)dx = pf(x)

We can conclude from property 2 that

d/dx∫pf(x)dx = pd/dx∫f(x)dx =pf(x)

• Property  5:

For a finite number of functions f₁, f₂,… f_n and a finite number of real numbers p₁, p₂,… p_n, the following formula can be used:

∫[p₁f₁(x) + p₂f₂(x)….+p_nf_n(x) ]dx = p₁∫f₁(x)dx +  p₂∫f₂(x)dx + ….. +  p_n∫f_n(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

Conclusion

An indefinite integral is an integral that does not have any upper or lower bounds and therefore has no upper or lower bounds.According to mathematics, if F(x) is an anti-derivative of f(x), then the greatest generic anti-derivative of f(x) is referred to as an indefinite integral and indicated by the symbol:∫f(x) dx = F(x) + C.The processes of differentiation and integration are diametrically opposed to one another. Two indefinite integrals with the same derivative lead to the same family of curves. It can be shown that the integral of the sum of two functions is equivalent to the sum of the integrals of the two functions that are presented.