Integration Made Easy

By Masood Amir

Calculus:

Calculus is a tool used to measure change or variation of a function with respect to the independent variable.

Differential Calculus:

It is the branch of calculus used to measure change or variation of a function in a very small interval of time, the techniques use to measure such changes is called “differentiation”.

Integral Calculus:

It is the branch of calculus used to measure changes or variation over an interval of independent variable, e.g to find length of curve, the area of region and the volume of a solid in a specified period of time.

The technique used to measure such changes or variation is called “Integration” or “Antiderivatives”. It a reverse process of differentiation.

Mathematically, Integration is defined as “ If f’(x) represents the differential coefficient of f(x), then the problem of integration is given f’(x), find f(x) or given dy/dx, find y.

Notation:  ”∫ ” is used to show the integration, it is a symbol of “S” derived from the word “Sum”. i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.

Techniques of Integration:

As we know that integration is the reverse process of differentiation, our problem is to find the function f(x) or Y, when f’(X) or dy/dx is given.

dy/dx = f’(X)

∫dy = ∫f ’(X)dx

Y= f(x) is our solution

Ist Formula of Integration (1st Rule of Integration)

Indefinite Integration:

Ist Formula of Integration (Ist Rule of Integration):

 If  y = xn

∫ dy = ∫ xndx

y = xn+1/n+1  + C

Why “C”:

In the process of differentiation, we eliminate constant, as the derivative of a constant is “zero”.

So, In functions like Xn, Xn+ 6, Xn + 3 ,  Xn – K, the derivatives of all of them is Xn-1, in finding the anti derivative of  Xn-1, we put a constant “C”, as we don’t know which constant was present in the original function, and this can be found If we have initial boundary values (Definite Integral).

Example: Solve ∫x3dx

Solution:        x3+1/3+1 + C   = x4/4 + C

Example: Solve ∫(x3 + x2 + 5x + 6)dx

Solution:      ∫x3dx + ∫x2dx + ∫5xdx + ∫6dx

                         X4/4 + x3/3 + 5x2/2+ 6x + C

 

Worksheet # 1

 

Find the Integral of the following:

 

SET 1

a. ∫(x3-4x2+5x-6)dx                       

  1. ∫(3x5-4x3+3x2)dx

c. ∫(ax5-bx4)dx                               

  1. ∫(x3/2-5x4/3+3x2)dx

e.∫(4√3x2 -2x)dx                     

  1. ∫(3x + 5x2 –x3/2-0.4x4)dx

g. ∫(x(8x-1/2)dx                                       

  1. ∫(2-x)(4+3x)dx
  2. ∫(x-3 + x-4)dx                               
  3. ∫(2x3-3)3x4dx

k. ∫(4x7 + 3x12 -5x8 + 2x -1)dx            

  1. ∫(ax3 – bx2 + cx –d)dx
  2. ∫(1/x3 + 2/x2 -6)dx
  3. ∫(-3x-8 + 2√x)dx
  4. ∫((x3 -5)(2x + 5)dx
  5. ∫(7x-6 + 5√x)dx

 

 

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