Indefinite Integrals Introduction

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During the course of study of Mathematics, we must have come across several parts of inverse operations like (addition, subtraction) (multiplication, division) (forming an equation whose roots are given - solving a given equation) and so on.

In practical situations, we may be interested to know the position of an object at an instant, when velocity of the object at that instant is given. That is if s(t) is the dispalcement of an object in time t and we know .

How can we find the displacement at time t ?

How do we find the velocity of a moving object, when its acceleration and initial velocity is known?

These three problems involves the process of finding the function whose derivative is given.

Integration and differentiation are a pair of inverse operations. So far, from a given function, we have been finding its derivative but the question arises: what is the function whose derivative is known? If the derivative of a function is given, then the function itself is called anti-derivative or integral. We illustrate it with the help of an example.

Consider the function f(x) = x⁴ then its derivative is given by f^'(x) = 4x³. The question arises: given f^'(x) = 4x³ what is f(x)?