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):

Let ∫ 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 X^n-1, in finding the anti derivative of  X^n-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 

∫(x³-4x²+5x-6)dx                        ∫(3x⁵-4x³+3x²)dx

∫(ax⁵-bx⁴)dx                               ∫(^x3/2-5x^4/3+3x²)dx

∫(⁴√3x² -2x)dx                            ∫(3x + 5x² –x³/2-0.4x⁴)dx

∫(x(8x-1/2)dx                               ∫(2-x)(4+3x)dx

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

∫(4x⁷ + 3x¹² -5x⁸ + 2x -1)dx          ∫(ax³ – bx² + cx –d)dx

∫(1/x³ + 2/x² -6)dx                        ∫(-3x^-8 + 2√x)dx

∫((x³ -5)(2x + 5)dx                         ∫(7x^-6 + 5√x)dx