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.
Thanks for the information.
Integrate 7x^2
= (7x^3)/3 + c
= 7/3 x^3 + c
Please double-check.
Or did I make a mistake somewhere?
Dear Darrick_3658,
It is my careless mistake. It should be 7/3 x^3 + c.
Thanks.
Regards,
ahm97sic