Properties of Indefinite Integration

An indefinite integral is the integration of a given function having no boundaries. Integration is the antiderivative of the function, since it reverses the differentiation process. The indefinite integral is a fundamental concept in calculus, with different properties of indefinite integration, and applying limiting points to it converts it to definite integrals. 

As a result, an indefinite integral is defined as having neither an upper nor a lower limit. Antidifferentiation, also known as indefinite integration, reverses the differentiation process. Finding a function F such that F’ = f., where I is current in Amperes, C is capacitance in Farads, V is voltage in Volts, and t is time in seconds, given a function f.

Indefinite Integration: Meaning

Integration is a term used to describe the determination of the area contained by a curve concerning one of the coordinate axes. Integration by parts, substitution, partial fractions, and integration of inverse trigonometric functions are all approaches for solving indefinite integrals. Learn more about indefinite integrals, including key formulae, examples, and the distinction between indefinite and definite integrals. 

A differentiable function F whose derivative can be identical to the original function f is an antiderivative, primitive integral, primitive function, indefinite integral, or inverse derivative of a given function f in calculus. F’ = f is a symbol that can express this. 

Antidifferentiation (or indefinite integration) is how you can solve antiderivatives, whereas differentiation is the opposite operation – it is the process of solving a derivative. Upper case alphabets, for example, F & G, are frequently used to designate antiderivatives. Take a look at the properties of indefinite integration listed below.

Indefinite Integral Properties

First Property

d/dx ∫ f(x) dx = f(x) and ∫ f ‘(x) dx = f(x) + C,

where,

C could have any arbitrary value.

Proof

Allow G to be any anti-derivative of f. 

d/dx[G(x)] = f(x) ……. (1)

∫ f(x) dx = G(x) + C

d/dx[∫ f(x) dx] = d/dx[G(x) + C]

= d/dx[G(x)] [‘.’ d/dx (C) = 0]

= f(x) [from (1)]

Therefore, d/dx[∫ f(x)dx] = f(x)

We’re already familiar with this,

 d/dx[f(x)]

dx = ∫ f ‘ (x) dx

⇒ ∫ f ‘(x)dx = f(x) + C

Hence, C is referred to as the integration constant in this case. 

Second Property

Two indefinite integrals of the same derivative produce the same set of curves and are equal.

Proof

Let f and g be two functions with the property that,

d/dx ∫ f(x) dx = d/dx ∫ g(x)dx

⇒d/dx ∫ f(x)dx – d/dx ∫ g(x)dx = 0

⇒d/dx[∫f(x)dx – ∫g(x)dx] = 0

d/dx(Constant) = 0

⇒ ∫ f(x)dx – ∫ g(x) dx = 

Where C is an arbitrary constant

∫ f(x)dx = ∫ g(x) dx + C

If C = C2 – C1, then

∫ f(x)dx = ∫ g(x) + C2 – C1

⇒ ∫ f(x)dx + C1 = ∫ g(x) + C2

{∫ f(x)dx + C1,C2 ∈ R} and {∫ g(x)dx + C1,C2 ∈ R} are identical }

∫ f(x)dx and ∫ g(x)dx are equal

Third Property

Assuming f and g represent two main functions, then [f(x) + g(x)] is the result. 

dx = f(x) dx plus g(x) dx

d/dx ∫ f(x) dx = f(x)

d/dx ∫ [f(x) + g(x)] dx = f(x) + g(x) …….(1)

Consider d/dx [∫ f(x) dx + ∫ g(x) dx]

= d/dx ∫ f(x) dx + d/dx ∫ g(x) dx

= f(x) + g(x)…..(2)

 d/dx ∫ [f(x) + g(x)] dx = d/dx [∫ f(x) dx + ∫ g(x) dx]——- (3)

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Thus, the sum of the integrals of the functions equals the sum of the integrals of the functions.

These are a few properties of indefinite integration that will help you understand and solve the equations easily.

Dissimilarities Between Indefinite and Definite Integrals

• A definite integral has lower and upper bounds and yields a constant result when solved. An infinite integral is one in which no boundaries are imposed, and an arbitrary constant is introduced as a requirement.

• When the number’s top and lower boundaries are constant, the definite integral expresses it. An infinite integral is a generalisation of a family of functions with derivatives f.

• Although the values or solutions obtained from definite integrals are constant, they might be positive or negative. An indefinite integral solution is a general solution with a constant value appended to it, commonly symbolised as C.

• In a definite integral, the upper and lower bounds are always constant. Because it is a generic representation, there are no limitations for indefinite integrals.

Conclusion

An indefinite integral is a function that performs the antiderivative of another function. It can be shown as an integral symbol, function, and dx. The indefinite integral is a simpler approach to represent obtaining the antiderivative. In addition, the indefinite integral is related to but not the same as the definite integral.