Mathematics > Integrals

Add Comment Bookmark share + Refresher Material


  • 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 y-axis.
  • 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

   

   

   

  • Integration by substitution

   

  • Integrals of some special functions

   

   

  • Integration by Parts
  • For given functions f1 and f2, we have

   

  • 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

))

  

Comments Add Comment
Ask a Question