Indefinite Integral: Types, Properties and Proof

Indefinite integral is one of the most often used operations in Calculus. The two important operations in calculus are differentiation and integration. Integration is the process of discovering a function's derivative; on the other hand, differentiation is the opposite. 

• Integrals are a significant component of Integration. 
• Hence, the Indefinite integral is the integration of a function with no limits. 
• Integration and differentiation are the opposite or reverse processes.
• Integration is also referred to as the antiderivative of function. 
• Different methods of integration such as integration by parts, integration by substitution, integration of partial fractions, and integration of inverse trigonometric functions are used to solve indefinite integrals in Calculus.

Read More: Inverse Trigonometric Formula

Key Terms: Calculus, Indefinite Integral, Differentiation and Integration, Integral, function.

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What are Integrals?

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Assuming that a function f is differentiable in the interval I, meaning that each point in I has a copy of the derivative, f'. 

• Antiderivatives are functions that may have served as a derivative (or primitive). 
• The indefinite integral of the function is the name of the formula that generates all of these antiderivatives. 
• The search for antiderivatives is known as integration.

Read More: Complex Numbers and Quadratic Equations

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Types of Integrals 

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The two types of integrals that are typically used in mathematics for Calculus are:

1. Definite
2. indefinite integrals

• A definite integral indicates a number when both the bottom and upper boundaries are constants. 
• The indefinite integral represents a large family of functions, whose derivatives are f. 
• The basic difference between these integrals is the constant.

Read More: Geometric Mean (G.M.)

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What are Indefinite Integrals?

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If an integral has no upper or lower boundaries, it is said to be indefinite. The most generic anti-derivative of f(x) is known as an indefinite integral and denoted, in mathematics, by F(x), which is any anti-derivative of f(x).

∫f(x) dx = F(x) + C

The antiderivative integrals of functions are not all the same. 

• There are infinitely many antiderivatives of each of the different functions that may be produced if C is arbitrarily chosen from the set of real numbers. 
• For this reason, C is frequently referred to as an arbitrary constant. 
• The parameter C is used to obtain various antiderivatives (or integrals) of the supplied function.

Read More: Bayes Theorem Formula

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Indefinite Integrals vs Definite Integrals 

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A definite integral indicates a number when both the bottom and upper boundaries are constants. The indefinite integral represents a large family of functions, whose derivatives are f. 

• The basic difference between these integrals is the constant. 
• Hence, an indefinite integral is a function that is antiderivative of another function. 
• It can be visually represented as an integral, a symbol, a function, and as a dx at the end. 
• Although the definite integral and the indefinite integral are comparable, they are not the same.

Read More: Sequence and Series

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Indefinite Integrals Formulas

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Most commonly required formulas of Indefinite Integrals are provided below in the table. Each of them has different functions as per the operation.

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Properties of Indefinite Integrals

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Various properties of Indefinite Integrals are provided below in the table: 

Read More: Real Numbers Formula

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Proof for each Property of Indefinite Integrals 

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A few properties of indefinite integrals are– 

Property 1: Differentiation and integration are the exact opposites of one another.

\(\frac{d}{dx} \int f (x) dx = f (x)\)

Proof: Let f be a function f so that its anti-derivative is F, i.e.

\(\frac{d}{dx}F (x)= f (x)\)

\(\int f (x) dx = F (x) + C\)

After differentiation,

\(\frac{d}{dx} \int f (x) dx = \frac{d}{dx} (F (x) + C)\)

Since, zero is the derivative of any constant function. Hence,

\(\frac{d}{dx} \int f (x) dx = \frac{d}{dx} (F (x) + C)\)

= \(\frac{d}{dx} F (x)\)

= f (x)

f’ (x) = \(\frac{d}{dx} f (x)\)

Hence, proved:

∫ f ‘ (x) dx = f(x) + C

Property 2: Two indefinite integrals leading to the same family of curves are equal if they have the same derivative.

Proof: Consider f and g as two functions:

\(\frac{d}{dx} \int f (x) dx = \frac{d}{dx} \int g (x) dx\)

or

\(\frac{d}{dx}[ \int f (x) dx - \int g (x) dx] = 0\)

Then,

\(\int f (x) dx - \int g (x) dx = C\)

or 

\(\int f (x) dx = \int g (x) dx + C\)

Property 3: The integral of the sum of two functions is equal to the total of the integrals of the functions. 

\(\int [f (x) + g (x)] dx = \int f (x) dx + \int g (x) dx\)

Proof: Already known due to the property 1:

\(\frac{d}{dx} ]\int [f (x) + g (x)] dx] = f(x) + g(x)\) …..1

\(\frac{d}{dx} ]\int [f (x) + g (x)] dx] = \frac{d}{dx} \int f (x) dx + \frac{d}{dx} \int g(x) dx = f(x) + g(x) \) ….. 2

\(\int [f (x) + g (x)] dx = \int f (x) dx + \int g(x) dx\)

Property 4: No need to Proof.

Property 5: For any real value of p,

∫ pf(x) dx = p ∫ f(x) dx

Proof: Due to property 1,

\(\frac{d}{dx} \int f (x) dx = f (x)\)

Also, 

\(\frac{d}{dx}[p \int f(x) dx] = p \frac{d}{dx} \int f(x)dx = pf(x)\)

∫ pf(x) dx = p ∫ f(x) dx

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Solved Examples

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Problem 1: Evaluate 3ax/(b² + c²x²) dx

Solution: I = 3ax/(b² + c²x²) dx is the integral's evaluation formula.

Consider v = b² + c²x², therefore.

dv= 2c²x dx

As a result, 3ax/(b² + c²x²) dx

dv/v = (3ax/2c²x)

As a result of canceling x on the numerator and denominator, we now have

= (3a/2c²)∫dv/v

= (log(3a/2c²)) |b² + c²x²| + C

where C is a randomly chosen constant.

Also Read:

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Things to Remember

• Differentiation and integration are the exact opposites of one another.
• Two indefinite integrals leading to the same family of curves are equal if they have the same derivative.
• The integral of the sum of two functions is equal to the total of the integrals of the functions. 
• For a finite number of functions f₁, f₂…. f_n and the real numbers p₁, p₂…p_n, ∫[p₁f₁(x) + p₂f₂(x)….+p_nf_n(x) ]dx = p₁∫f₁(x)dx + p₂∫f₂(x)dx + ….. + p_n∫f_n(x)dx
• For any real value of p, ∫ pf(x) dx = p ∫ f(x) dx
• ∫ f(x) dx means Integral of f as per the function of x.
• f(x) in ∫ f(x) dx means Integral.
• x in ∫ f(x) dx means Variable of integration.
• An integral of f indicates a function F such that F′(x) = f (x).
• Integration refers to the process of finding the integral.
• Constant of Integration means any real number C that is considered as constant function.

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