Integration as an Inverse Process of Differentiation - Practice Questions & MCQ

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given. 

For example,

In the above example, the function cos(x) is the derivative of sin(x). We say that sin(x) is an anti-derivative (or an integral) of cos(x). Similarly, x² and e^x  are the anti derivatives (or integrals) of 2x and e^x respectively.  

Also note that the derivative of a constant  (C) is zero. So we can write the above examples as:

Thus, anti derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti-derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers. 

For this reason C is referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. 

If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx. 

By definition,

Here, 

Rules of integration

f(x) and g(x) are functions with antiderivatives ∫ f(x) and  ∫ g(x) dx. Then,

Standard Integration Formulae

Based on this definition and various standard formulas (which we studied in Limit, Continuity and Differentiability) we can obtain the following important integration formulae,