What is an integral and what is the purpose of integration?

Integration

Integration is an inverse function of derivatives. It is necessary to understand derivatives first before understanding antiderivative. A derivative is defined as the rate of change of its function value. It describes that the value of a function of a real variable describes the change with the change of its input value. Derivatives and antiderivatives are the fundamental tools of calculus. These are also called integral. Integrals are of two types:

  • Definite Integral
  • Indefinite integral

The term definite integrals is used for integrals having functions of lower and upper bound limits whereas indefinite integrals are known as antiderivatives. The symbol used for antiderivative is assembled with an English alphabet ‘S’ and represented as

It is also be represented as F’.

Integrals can be solved with the help of integral calculator by putting the expression in the text box available on the integral calculator tool. The mathematical steps which are involved during the solution of an antiderivative function can be better understand in the later part of this topic.

Definition of Integrals:

Antiderivative refers to the derivative of a function equal to its original function. It actually reverses the result which the derivative generates. A function has more than one antiderivative which is separated by a constant. The process of finding all derivatives of a function is known as anti-differentiation

 =  + c

Purpose of Integration:

As we know integrals are used to find the displacement, area, volume etc. A list of formulas are given below to evaluate a function. These are also called the properties of antiderivatives.

  • i).
  • ii). dx = x + c
  • x dx = – cos x + c
  • iv). x dx = – cot x + c
  • v). x (cot x) dx = –  cosec x + c
  • vi). x dx = sin x + c
  • x dx = tan x + c
  • x (tan x) dx = sec x + c
  • ix). x dx = ln IxI + c
  • x). dx = ex + c
  • xi). x dx = (ax/ln a) + c (where a ≠ 1, a >0)
  • n dx = ((xn+1) / n+1)) + c (where n ≠ 1) (Also called Power Rule)

Antiderivative can be find out by using the above formulas in which a step by step calculation is required. Whereas, to skip the mathematical calculations, an antiderivative calculator can be used to get the result with steps.

Applications of anti-derivatives:

Find the antiderivative of f (x6)

Solution:

6 dx

We know the formula

n dx = ((xn + 1) / n – 1)) + c

Substitute n=6

6 dx =  + c

6 dx =  + c


Example 2:

Evaluate 2 – 14x + 4cosx)dx

Solution:

(12x2 – 14x + 4cosx) dx

= 2 dx –  dx +  dx

= 12 2 dx –  dx + 4  dx

Solve it

I = 12 .  – 14 .  + 4 . sinx + C

= 4x3 – 7x2 + 4.sinx + C


Example 3:

Evaluate 3 – 11x2 + 5x + 6)dx

Solution:

Given expressions is 3 – 11x2 + 5x + 6)dx

In the given expressions, there are four terms which can be separately integrated.

3 – 11x2 + 5x + 6)dx  = 3 dx – 2 dx +  dx + dx

=7 3 dx – 11 2 dx +  dx + dx

= 7 ( ) – 11 ( ) + 5 ( ) + 6 (x) + C

=   x4  –   x3 +   x2 + 6x + C


Example 4:

8 – 21x6 + 3x2 + 7) dx

Solution:

Given  is 8 – 21x6 + 3x2 + 7) dx

We Can integrate each term separately

8 – 21x6 + 3x2 + 7) dx = 8 dx – 6 dx + 2 dx +  dx

=9 8 dx – 6 dx +3 2 dx +7  dx

=9 ( ) – 21 ( ) + 3 ( ) + 7(x) + C

After Simplification

= x9 – 3x7 + x3 + 7x + C


 Example 5:

 dx

Solution:

 dx  –  + dx

= 7  + dx

= 7  + dx

= 6  –  + 12lnIxI

= ¾ x8 – x2 + 12lnIxI + C


Example 6:

6 + 2z4 – z2 dz

Solution:

6 + 2z4 – z2 dz

Integrate the terms separately

6 + 2z4 – z2 dz = 6 + 4 2

=  + 4 2

=  +  

=  z7 + z5  z3  + C

The basic concept of antiderivatives has been explored with the help of different examples. Students have to grip different integral properties to be successful in the future.