Is One or Both of the Integral Solutions Correct?

[DocZaius]

Consider this indefinite integral:

\int(x^2+6)(2x)dx

There are two ways I could approach solving it. The first one would be to multiply the terms, then solve using the sum rule. That approach would yield this solution:

\frac{x^4+12x^2}{2} + C

The other way would be by substitution. It would yield this:

\frac{(x^2+6)^2}{2} + C

The second constant is 18 less than the first one. I realize that since they're arbitrary constants, it might not matter, but I want to be clear. My question is:

Strictly speaking, are both solutions equally correct or is only one correct? If it's neither completely equal or one being the only correct answer, and it's merely a matter of preference, could you elaborate in what context which solution is preferable?

Thanks!

 

[Bohrok]

Both are equal since C is just an arbitrary constant, although one might be more "correct" than the other depending on what method of integration you are supposed to do and how the result might be expected to look like. In some problems where you have a number and the constant of integration, you can combine the two. For example, you might have 18 + C₁, then let C = 18 + C₁, so you have just C for the constant, which makes some integrals look cleaner.

 

[tiny-tim]

Yes, both solutions are correct,

but the second (chain rule) one is neater, since you can instantly both do it and check it.