
Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation.

Let. Then we write These integrals are called indefinite integrals or general integrals; C is called constant of integration. All these integrals differ by a constant.

From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the yaxis.

Some properties of indefinite integrals are as follows: . For any real number k,

More generally, if
are functions and
are real numbers. Then

More Integral Formulae

Integration by partial fractions

Integrals of some special functions

Integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}. Care must be taken in choosing the first function and the second function. Obviously, we must take that function as the second function whose integral is well known to us.
Examples
Question
Find the integral of
Solution
=
Put t = cos x so that dt = – sin x dx
Therefore,
=
Question
Evaluate
Solution
Hence
x – 3 = t
dx = dt
))