Hi Eagle
The difference between the two methods is something like there
are two ball bearings, one large and one small.
There are two pipes, one large and one small.
Case 1 (Indefinite
Integration Method)
Only the large ball bearing passes through a large pipe (the
small ball bearing is assumed to be stucked and it does not pass
through the small pipe) and the large ball bearing is being melted
when it passes through the pipe. Indefinite integration is
something like with the melted iron, we will remould it into the
large ball bearing. We know that there is the small ball bearing
but it is assumed to be stucked and so there is no melted iron of
the small bearing, so we will not be able to remould the small ball
bearing but we know that we have a small ball bearing at
first, so we have to add the constant c in the indefinite
integration method.
Case 2 (Reversal
Method)
Both ball bearings pass through the pipes and they are being
melted when they pass through the pipes. Reversal Method is
something like with the melted iron, we will be able to remould the
large and small ball bearings ie an exact reversal, so there is no
need to add the constant c in the reversal method.
Example 1
A typical question
that requires
A typical question that
requires the use of
the use of reversal
method
is
indefinite integration
method
(1) Given that y =
2x^3 + 5 (1) y
function is not given ie y = 2x^3 + 5 is
(2) Differentiate y = 2x^3 +
5 not
given
(3) Next, integrate
6x^2
(2) Integrate 6x^2
Reversal
Method
Indefinte Integration
Method
(1) Given y = 2x^3 +
5 (1)
y = 2x^3 + 5 is not given
dy/dx =
6x^2
(2) Next, integrate
6x^2 (2)
Integrate
6x^2
(use integrate dy/dx =
y)
= (6x^3)/2 + c
= 2x^3 +
5 =
2x^3 + c
Important Note :
Mathematically, when we integrate
6x^2 = 2x^3 but in the case of the reversal method, we are given
the information that the original y has the constant 5, so the
answer is 2x^3 + 5. However, in the case of indefinite integration
method, there is no given information that the original y has the
constant 5 since the original y is not given, so the constant
c is used to represent the 5 in the original y in the answer ie
2x^3 + c.
Please take another look at the two methods.
The reversal method is able to get back the constant 5 because
the original y is given while the indefinte integration method is
unable to get the constant 5 because the original y is not given
and hence a constant c is used to represent the possible constant
as in the case of indefinite integral.
Example 2
Reversal
Method
Indefinte Integration
Method
(1) Given y = 2x^3 +
5 (1)
y = 2x^3 + 5 is not given
dy/dx =
6x^2
(2) Next,
integrate 7x^2
(2) Integrate 7x^2
= 7/6 integrate
{6x^2}
= (7x^2)/3 + c
(use integrate dy/dx =
y)
= 7/3 x^2 + c
= 7/6 {2x^3 + 5}
= 7/3 x^2 + 35/6
The reversal method is able to give the exact constant 35/6
because the original y is given while the indefinte integration
method is unable to get the constant 35/6 because the original
y is not given and hence a constant c is used to represent the
possible constant.
Thank you for your kind attention.
Regards,
ahm97sic
