Page 1
INTEGRATION
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc.
whenever the ratio of change of a quantity is known we can always calculate the total change in a
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments
of force etc. are some of the innumerable applications of integration.
Page 2
INTEGRATION
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc.
whenever the ratio of change of a quantity is known we can always calculate the total change in a
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments
of force etc. are some of the innumerable applications of integration.
INTEGRALS AS AN ANTIDERIVATIVE
The process of finding the antiderivative is called integration.
Integration is the inverse process of differentiation.
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ??
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? )
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? )
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand.
So ??? (?? )???? is not unique due to presence of ' ?? '.
Page 3
INTEGRATION
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc.
whenever the ratio of change of a quantity is known we can always calculate the total change in a
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments
of force etc. are some of the innumerable applications of integration.
INTEGRALS AS AN ANTIDERIVATIVE
The process of finding the antiderivative is called integration.
Integration is the inverse process of differentiation.
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ??
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? )
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? )
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand.
So ??? (?? )???? is not unique due to presence of ' ?? '.
Geometrical Significance
The derivative of a function give the slope whereas, the integration of a bounded function gives a
particular algebraic area.
Page 4
INTEGRATION
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc.
whenever the ratio of change of a quantity is known we can always calculate the total change in a
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments
of force etc. are some of the innumerable applications of integration.
INTEGRALS AS AN ANTIDERIVATIVE
The process of finding the antiderivative is called integration.
Integration is the inverse process of differentiation.
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ??
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? )
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? )
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand.
So ??? (?? )???? is not unique due to presence of ' ?? '.
Geometrical Significance
The derivative of a function give the slope whereas, the integration of a bounded function gives a
particular algebraic area.
Derivative of Integral, Integral of Derivative
?? ????
? ?? (?? )???? = ?? (?? )
? {
?? ????
(?? (?? ))}???? = ?? (?? ) + ?? ? { ?? (?? )?? '
(?? ) + ?? '
(?? )?? (?? )} ???? = ?? (?? ) · ?? (?? ) + ?? ? {
?? '
(?? ) · ?? (?? ) - ?? (?? )?? '
(?? )
{?? (?? )}
2
} ???? =
?? (?? )
?? (?? )
+ ?? ? ?? '
(?? (?? ))?? '
(?? )???? = ?????? (?? ) + ??
Page 5
INTEGRATION
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc.
whenever the ratio of change of a quantity is known we can always calculate the total change in a
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments
of force etc. are some of the innumerable applications of integration.
INTEGRALS AS AN ANTIDERIVATIVE
The process of finding the antiderivative is called integration.
Integration is the inverse process of differentiation.
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ??
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? )
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? )
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand.
So ??? (?? )???? is not unique due to presence of ' ?? '.
Geometrical Significance
The derivative of a function give the slope whereas, the integration of a bounded function gives a
particular algebraic area.
Derivative of Integral, Integral of Derivative
?? ????
? ?? (?? )???? = ?? (?? )
? {
?? ????
(?? (?? ))}???? = ?? (?? ) + ?? ? { ?? (?? )?? '
(?? ) + ?? '
(?? )?? (?? )} ???? = ?? (?? ) · ?? (?? ) + ?? ? {
?? '
(?? ) · ?? (?? ) - ?? (?? )?? '
(?? )
{?? (?? )}
2
} ???? =
?? (?? )
?? (?? )
+ ?? ? ?? '
(?? (?? ))?? '
(?? )???? = ?????? (?? ) + ??
Fundamental rules of Integration
? { ?? 1
(?? ) ± ?? 2
(?? ) ± ?? 3
(?? ) ± ? ± ?? ?? (?? )} ???? = ? ?? 1
(?? )???? ± ? ?? 2
(?? )???? ± ? ?? 3
(?? )???? ± ? ± ? ?? ?? (?? )????
? ???? (?? )???? = ?? ? ?? (?? )???? , where ?? is a constant (?? ? 0)
? ?? '
(???? + ?? )???? =
?? (???? + ?? )
?? + ?? (?? ? 0)
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