Indefinite Integrals

The problem of defining and calculating the area of the region bounded by a graph of a function has motivated to the discovery of integral calculus. If the function f is differentiable in an interval, then f' is its derivative. In order to determine the function, when f' is given is a question.

The formula that gives all these anti-derivatives is called the indefinite integral of the function and such process of finding anti-derivatives is called integration. For example, if the velocity of a particle at a given instant is known, in order to determine the position of the object at any instant, the process of integration is useful. Integration is the reverse proce3ss of differentiation. If we are given the derivatives of a function and asked to find its primitive, the process we use is called indefinite integral. Also note that d(sin x)/dx = cos x and also d[(sin x) +a]/dx = cos x. Hence, we see that while integrating, the anti-derivative of the above function is not unique. There exist infinitely many anti-derivatives of each of these functions which can be obtained by choosing c arbitrarily from the set of real numbers. c is referred to as arbitrary constant. By writing we mean the indefinite integral of f with respect to x. If f(x) = 2x , then =x² +C. For different values of C we get different integrals. We see that y =x² +C , represents a family of curves. By substituting different values for C, we get different members of the family. This together forms the indefinite integral. Hence, the geometrical interpretation of indefinite integral is that we get a family of similar curves obtained by shifting one of the curves parallel to it.