∫ (2x-1)^2 dx could have two answers:
a) ((2x-1)^3)/6 + C
b) 4/3x^3 - 2x^2 + x + C
Now the question is, how are the two answers related?
I attempted to answer the question by:
"The two answers are related because when finding the derivative of those two different answers, they both result to the original equation. Therefore, they are related by some constant C"?
How correct is that explanation? Or am I missing out something for my answer?
Any help is greatly appreciated!
since
((2x-1)^3)/6 = 4/3 x^3 - 2x^2 + x - 1/6
the two solutions differ by 1/6
Since C is an arbitrary constant, its value does not matter.
Your explanation is partially correct, but there is a bit more to it. To correctly explain how the two answers are related, you need to consider the constant of integration, denoted by "C" in both answers.
When you perform indefinite integration, you typically add the constant of integration to account for all possible antiderivatives. In this case, both the answers have the constant of integration "C" at the end.
The two answers, ((2x-1)^3)/6 + C and 4/3x^3 - 2x^2 + x + C, differ by a polynomial term. Specifically, ((2x-1)^3)/6 + C can be expanded to (4x^3 - 6x^2 + 3x - 1)/6 + C, while 4/3x^3 - 2x^2 + x + C can be written as (4x^3)/3 - 2x^2 + x + C.
If we compare these two expressions, we can see that they differ by (4x^3 - 6x^2 + 3x - 1)/6. This additional term is what causes the difference between the two answers. However, since we are dealing with indefinite integration, any constant term could be absorbed into the constant of integration "C". Therefore, the two answers are indeed related by some constant, which includes the polynomial term mentioned above.
In summary, your explanation is mostly correct, but you should explicitly state that the two answers differ by a polynomial term, which is permissible in indefinite integration and can be absorbed into the constant of integration.