In Calculus, integration is a process that involves the computation of either a definite integral or an indefinite integral. The development of integral calculus has arisen in order to solve the following two different types of problems, such as the problem of finding the function if the derivatives are given and the problem of finding the area bounded by the graph of the function under certain conditions. These two problems lead to the formation of integral calculus, which has two forms of integrals such as definite integrals and indefinite integrals. Here, we will discuss in detail why integration is considered as an inverse process of differentiation.
Integration as an Inverse Process of Differentiation – Reason
We know that differentiation is the process of finding the derivative of a function. Whereas integration is the process of finding the antiderivative of a function. Hence, we can say that integration is the inverse process of differentiation. In the integration process, instead of differentiating a function, we are provided with the derivative of a function and asked to find the original function (i.e) primitive function. Such a process is called anti-differentiation or integration.
Consider an example,
d/dx (x^{3}/3) = x^{2}
Here, x^{3}/3 is the antiderivative of x^{2}.
Actually, there exists infinitely many integrals or antiderivatives of each function, which can be obtained by choosing the arbitrary constant C from the set of real numbers.
(i.e) (d/dx) [(x^{3}/3)+ C] = x^{2}.
Hence, C is a parameter where one gets different antiderivatives (or) the integrals for the given function.
Therefore, in general,
“If there is a function F, such that (d/dx) F(x) = f(x), for all x belongs to the interval I, then for any arbitrary real number C is called the constant of integration.
(i.e) (d/dx) [F(x)+C] = f(x) , x ∈ I
Hence, {F+C, C∈R} represents the family of antiderivatives of the function “f”.
The notation ∫ f(x) dx represents the entire class of antiderivatives and it can be read as the indefinite integral of the function f with respect to the variable x.
Therefore, symbolically we can write the integration process as:
∫ f(x) dx = F(x) + C
Note: dy/dx = f(x) and hence y=∫ f(x) dx.
Also, read: |
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Integration Phrases/Terms and their Meaning
x in ∫f(x) dx – Variable of Integration
∫ f(x) dx – Integral of function f with respect to x.
f(x) in ∫f(x) dx – Integrand
Integrate – Find the integral
Integration – The process of finding integral
C – Constant of integration. Any real number C is considered as a constant function.
An integral of f – A function F, such that F’(x) = f(x).
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