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^3)/3 + c
(use integrate dy/dx =
y)
= 7/3 x^3 + c
= 7/6 {2x^3 + 5}
= 7/3 x^3 + 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.